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THE ON CLASSIFICATION OF SEVENTH DEGREE I -CURVES WITH THE MAXIMUM NUMBER OF POINTS OF INTLRSECTION OF THE ODD BRANCH WITH A LINE by DANIEL ERIC SMITH, B.S. A THESIS IN MATHEMATICS Submitted to the Graduate Faculty of Texas Tech University in Pardal Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE Approved December, 2001 ,y^ A/Vi/V'^ ^ ACKNOWLEDGMENTS -^^,^ / ;^ ^ .,- \ I would like to thank several people on a professional level: Dr. Anatoly Korchagin. for introducing me to this field of studv and for having the patience and willingness to explain things multiple times when I needed it: Dr. David Weinberg and Dr. Wavne Lewis, for always taking the time to answer questions; Dr. Rodger Barnard, for taking the time to talk to me about mathematics and a career in mathematics; Dr. Harold Bennett, for the opportunity to continue my studies: Dr. Steve White, for both the challenges and a better view of mathematics: Dr. Jeff Dodd, for the advice and support. I would also like to thank all those that helped me in my education: Dr. Jerald .\bercrombie. Dr. Dem])sey, Dr. Pu-Sen Yeh, Dr. James Stagliano, Dr. Jimmy Chen, and last but certainly not least Terrv Marbut. On a personal level, there are many people who I am indebted to: Doug and Linda Smith who gave me life and have always supported me, Doug "Deuce" Smith, Steph Smith. Glenn Williams, Church Alickel (for introducing me to Dr. Korchagin. as well as being a great friend). Keith Nabb (a.k.a.. Xobb) David Martin, Jeff Garza, Joey Severino, Doug Meador. and Jeff Robinson. Finally I must thank Norma Aguirre, Shiela Muse, and Margaret Plunket for keeping the department running, and David Hensley for keeping the comput(ns running and helping us to have good machines to work on. 11 CONTENTS .ACKNOWLEDGMENTS ABSTRACT LIST OF TABLES LIST OF FIGURES I INTRODUCTION 1.1 Short introduction to \'iro's Method 1.1.1 The Toric Group 1.1.2 Space of Polynomials and Newton Polygons 1.1.3 \ l r o Patchworking II ENUMERATION AND RESTRICTION 2.1 2.2 Corollaries from Bezout and Harnack theorems Corollaries from the Gudkov-Rohklin congruence and its generalization 2.3 Nonprohibited cases ii iv v ix 1 3 3 6 9 14 14 15 19 35 35 44 49 59 69 74 III CONSTRUCTION 3.1 3.2 Construction using Viro"s Method Construction of Korchagin's cur\'es 3.2.1 Construction using Korchagin's curves 3.3 Harnack's Method: AZ-curves of degree 6 I\^ CONCLUSION BIBLIOGRAPHY 111 ABSTRACT We will start the classification of 7^'^ degree AZ-curves with the maximum number of points of intersections of the odd branch with a line. This is done in three steps. First, we enumerate all possible curves with the maximum number of points of intersections of the odd branch with a line. Next, we find some restrictions to those curves. Finally, we start the construction of curves with maximum number of points of intersection. IV LIST OF TABLES 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 4.1 4.2 4.3 4.4 4.5 Cases of arrangements for A Cases of arrangements for B Cases of arrangements for C Cases of arrangements ior D Cases of arrangments ior E Total number of cases Figure 3.28 A, B, and C Table of constructed curves from Figure (3.42) Constructed curves of isotopy type .4 Constructed curves of isotopy type B Constructed curves of isotopy type C Constructed curves of isotopy tyi)e D Constructed curves of isotopy tvpe E 20 23 25 30 33 34 49 68 70 71 72 73 73 \' LIST OF FIGURES 1.1 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 The curve/(re,?/,2;) = a;-I-y-I-2; and regenerations Isotopy pairs ( I P ^ , J u R C i ) Isotopy pairs (IP^ , J U RCi ) with signed regions Constructed curve ^3(12,1,2) A prohibited arrangement for ^ CgUCa ^(a,^);^(a,A7),^(a,/?,7),^4(a,/3,7) ^(a,/?,7,(5),A6(a,A7,'^),^7(Q;,/3,7,^)A(Q^,^,7,^) Ag(a,^)^o( ,/?,7) Au{a,p,j,S)Au{a,^,'y) 12 14 16 17 18 18 19 19 19 20 20 An{a,^,y,S),Au{a,l3,j,6), 2.10 B,{aJ),B2{aJ)A{a,^) 2.11 54(0;, /?,7),^5( , 13,7),Be{a,/5,7), 2.12 S i o ( a , ^ , 7 , ( 5 ) , . . . , B i 5 ( a , ^ , 7 , ( 5 ) 2.13 B i 6 ( a , ^ ) ; B i 7 ( a , ^ , 7 ) A 8 ( , A 7 ) 2.14 Bi9(a,/3);B2o(c.,^,7),52i(c^, A 7), 5 2 2 ( ^ , ^ 7 ) 2.15 B23(a,p,j,6),...,B2s{a,P,yJ) 2.16 B29(c.,A7) 2.17 Ci(Q^,/3),C2(a,^);C3(a,/3,7),C4(a,A7);C'5(a,^,7,^) 2.18 C6(a,/3,7),C7(a,/?,7),C8(a,^,7),C9(Q',^,7) 2.19 Cio(a,;^,7,(5),Cii(a,^,7,^) 2.20 Ci2(a,/3);Ci3(a,/3,7),Ci4(a,^,7);Ci5(a,^,7,c^),Ci6(a,/?,7,^) 2.21 D i ( a , ^ ) , L > 2 ( a , / 3 ) , A ( a , ^ ) , D 4 ( a , ^ ) , D 5 ( a , / 3 ) 2.22 D6(a,/3,7),...,A5(Q^,/?,7) 2.23 Di6(a,/3,7,(5),...,D25(a,/3,7,^) 2.24 D26(a,^,7),^27(a,/?,7),A8(cv,^,7),^29( ,/3,7) 2.25 D 3 o ( a , ^ , 7 , ^ ) , - . . , A 5 ( a , ^ , 7 , ^ ) Bjia,^, j),Bsia, /?, 7),B,{a^ p, 7) . 21 21 21 21 22 22 23 24 24 24 25 26 26 26 27 VI 2.26 Ds6{a,0,j,5,e),Ds7{a,l3,j,6,e),D^s{cy,p,^,6,s),Ds9{a,p,-f,S,e) 2.27 D4o(a,^,7),Ai(Q^,A7),A2(a,A7),A3(a,/?,7) 2.28 Du(a,^,^,S),...,D^9(a,^,^,d) 2.29 Dso{a,p,^,6,6),D^i{aJ,^,6,s),Ds2(a,/3,j,S,e),D53{a,l3,^,S,s) 2.30 D5A(a,P) 2.31 Ds5{a,l3,j),Dse(a,p,j),Ds7ia,P,j),D,s{aJ,j),D,9{a,(3,j) 2.32 Deo{a,p,J,S),...,De9{a, 13,^,6) . . 27 27 28 . . . . . .... 28 28 29 29 31 32 32 2.33 E,(a,(3),E2{a,^);Esia,^n),E,(a,^.^)-Esia,^,7,6) 2.34 Ee{a,l3,j),...,En{a,l3,^) 2.35 Eu{a,^,j,6),...,Eu{a,l3,^,6) 2.36 ;i8(a,/3,7,^, ), ^i9(a,^,7,(^,e);^2o(c^,^);^2i(a,/3,7),^22(a,^,7) 2.37 E23(a,/3,7,^),^24(c^,/?,7,^) 3.1 3.2 3.3 The curve x = y^ and the transformation hy~^ Third-degree curves with three points of intersection Fourth-degree curves, maximum number of ovals, planar point at (1 : 0 : 0), 4 points of intersection 3.4 Fourth-degree curves, maximum number of ovals, planar point at (1 : 0 : 0), 3 points of intersection 3.5 3.6 3.7 3.8 3.9 32 33 35 36 36 37 Fourth degree curve, planar point at (1 : 0 : 0), four points of intersection 37 Fourth degree curve, planar point at (1 : 0 : 0), three points of intersection 37 Figure 3.5 charts transformed by r+_ o ha Figure 3.6 charts transformed by hb Glued charts from Figures 3.7 and 3.8 38 38 38 39 40 40 41 41 3.10 Fourth-degree curves with four points of intersection 3.11 7*^ degree M-curve by gluing (3.9.11) with (3.10.1-4) and (3.2.1-2) . . 3.12 7*^ degree M-curve by gluing (3.9.12) with (3.10.1-4) and (3.2.3-4) . . 3.13 7^^ degree AZ-curve by gluing (3.9.21) with (3.10.1-4) and (3.2.1-2) . . 3.14 7*^ degree AZ-curve by gluing (3.9.22) with (3.10.1-4) and (3.2.3-4) . . vii 3.15 7^^ degree M-curve by gluing (3.9.31) with (3.10.5-8) and (3.2.1-2) . . 3.16 7*^ degree M-curve by gluing (3.9.32) with (3.10.5-8) and (3.2.3-4) . . 3.17 7^^ degree M-curve by gluing (3.9.41) with (3.10.5-8) and (3.2.1-2) . . 3.18 7^'' degree M-curve by gluing (3.9.42) with (3.10.5-8) and (3.2.3-4) . . 3.19 Conic C2 3.20 Conic C2 with construction curves 3.21 Curve C4 with construction curves 3.22 Curves C5I and C | 3.23 Curves Cl'\Cl'\Cl'\ 3.24 Conic C2 3.25 Curve C4 with construction curves 3.26 Curves Qi and C | 3.27 Curves Cl'\Cl'\C^'\ and C^'^ and Q^'^ 42 42 43 43 44 45 45 46 46 47 47 48 48 49 50 52 53 53 54 55 55 56 56 57 57 58 58 59 60 3.28 X21 singularity with the smoothings 3.29 Singularities AQ, Ai, As, DQ, Jio, Zi^, and X21 3.30 Charts of C^'^ 3.31 Charts of C^'^ 3.32 Charts of smoothings 3.33 Transformed charts of smoothings 3.34 Constructed from 3.30 and 3.33 3.35 Constructed from 3.30 and 3.33 3.36 Constructed from 3.30 and 3.33 3.37 Constructed from 3.30 and 3.33 3.38 Constructed from 3.31 and 3.33 3.39 Constructed from 3.31 and 3.33 3.40 Constructed from 3.31 and 3.33 3.41 Constructed from 3.31 and 3.33 3.42 Some AZ-curves of degree six 3.43 Seventh-degree M-curve i4io(ll, 1,2) constructed from Figure 3.42-1. Vlll 3.44 Seventh-degree AZ-curve A (12, 3) constructed from Figure 3.42-2. . . 3.45 Seventh-degree AZ-curve .4 o(7. 5.2) constructed from Figure 3.42-3. . 3.46 Seventh-degree AZ-curve .4 (8, 7) constructed from Figure 3.42-4. . . . 3.47 Seventh-degree M-curve A (7,3,4) constructed from Figure 3.42-5. . 3.48 Seventh-degree AZ-curve .4 (7,1,6) constructed from Figure 3.42-6. . 3.49 Seventh-degree M-curve A (11,1. 2) constructed from Figure 3.42-7 3.50 Seventh-degree AZ-curve A (3, 5, 6) constructed from Figure 3.42-8. . 3.51 Seventh-degree AZ-curve A o(3,2,9) constructed from Figure 3.42-9. . 3.52 Seventh-degree AZ-curve .4 o(ll, 1, 2) constructed from Figure 3.42-14. 3.53 Seventh-degree AZ-curve .4 o(7, 5. 2) constructed from Figure 3.42-15. 3.54 Seventh-degree AZ-curve .4 o(7, 4.3) constructed from Figure 3.42-16. 3.55 Seventh-degree AZ-curve .4 o(3,8,3) constructed from Figure 3.42-17. 3.56 Seventh-degree AZ-curve A o(3,1,10) constructed from Figure 3.42-18. 3.57 Seventh-degree AZ-curve A o(7,1,6) constructed from Figure 3.42-19. 3.58 Seventh-degree AZ-curve .4 4(11,1,2) constructed from Figure 3.42-30. 3.59 Seventh-degree AZ-curve .4 4(7,1, 6) constructed from Figure 3.42-31. 4.1 Isotoi)y pairs for degree six AZ-curves 60 61 61 62 62 63 63 64 64 65 65 66 66 67 67 68 69 IX CHAPTER I INTRODUCTION A real homogeneous algebraic plane curve of degree c/ is a real homogeneous polynomial f{x. y, z) of degree d with real coefficients, considered up to constant factors. The set R / = {{x : y : z) e MP^\f = 0} is called the set of real points of the curve and C / = {(x : y : z) ^ CP^ | / = 0} the set of complex points of the curve. .\ curve is said to be nonsingular if the system /; = o V (1.1) -^ has no non-zero solution. If the system (1.1) has a non-zero solution the curve is said to be singular and the solution is said to be a singular point of the curve. All curv(>s in this introduction will be considered nonsingular. unless otherwise noted. If a curve / is nonsingular then each connected component of R / is homeomorphic to a circle. There are two kinds of such circles. One kind separates the plane RP^ into two pi(>ces. one homeomorphic to a disk and one homeomorphic to the Mobius band. It is called an oval. The other kind of a connected component is called an odd branch and does not separate RP'^. The maximum number of components of a cur\e of degree (I is M = h i . In 1876 Harnack [3] proved that a curve has at most M components, he also proved that for each degree d there exists a curve with M components. .\ curve R / on RP^ is called an AZ-curve if it has the maximum number of components. If the degree of the curve is even then all the components are o\als. Suppose that a M-curve has more than one odd branch, then thev must have at l(>ast one point of intersection. This point of intersection must be a singular j)oint of the curve. This contradicts the fact that the curve is a AZ-curv(^ and is therefore nonsingular. So if the degree of a nonsingular projcntive curvc^ is odd, then the curve has (exactly one odd branch and all other components are ovals. 1 We will use the following coding scheme devised by Viro [15]. The odd branch is coded by (J) and an oval by (1). Let (.4) be some collection of ovals, then (1(^4)) means that one oval surrounds the collection (.4). If (.4) and {B) are two collections of ovals such that tliey are disjoint and no oval of one collection surrounds an oval of the other collection then we denote their union as (.4 II B). The following abbreviations are also used: AIL -M A = nA where the number of terms .4 in the disjoint sum is //.; n empty ovals n x 1 = n; and 1(1(... 1{A)...)) repetitions of the fragment 1( is n. = 1"{.4) where the number of We want to start the topological classification of seventh degree M-curves in the real projective plane having the property that their odd branch has the maximum number of points of intersection with a line. This is done by the topological classification of the pairs (RP^RCy U R C i ) (1.2) up to homeomorphisms under the following three conditions (where RCi and RC7 are curvets of degree one and seven respectively): 1. C7 is an M-curve, i.e. RC7 consists of one odd branch and 15 ovals. 2. The curves RC7 and RCi are in general position, i.e. int(ns(H-tions. 3. The curve RCi intersects the curve RC7 at seven points all lying on tli(^ odd branch. The solution of classification problems of this type are split into three steps. Enumeration of all admissible topological models of pairs (1.2) that satisfy the conditions 1-3. There are two kinds of such models: ones which contain algebraic repr(^sentativ(\s under consideration and those which do not. Restriction on the topological models. This is the proof that models ar(> not r(vilizable by algebraic curv(>s. SOIIK^ they have traiis\'ersal of the topological Construction of the algebraic representatives for the rest of the models. 1.1 Short introduction to Viros Method One method of constructing real algebraic curves is to use \'iro"s [15] method which takes a curve with singularities and smoothes them (see also [13, 14, 18, 19]). We introduce Viro's m e t h o d of gluing polynomial charts or briefly Viro's m e t h o d , following the paper by Korchagin [8]. 1.1.1 The Toric Group Let CP^ be the complex projective plane and (x : y : z) he the homogeneous point coordinate in CP^ . Also let (x, y) be the affine coordinates in the affine plane C^ = (CP'^ \ \^z = 0}. U '^[,ip2^^3 are homogeneous polynomials of the same degree and have no common factors, then we define a projective transformation (p b>' the formula (p{x : y : z) = {cpi{x,y, z) : (/?2(3^, ?/,'-^) : ^si^^y, z)). We define the restriction of (fix -.y.z) to the affine chart CP^ \ (z = 0} by ipix, y) = (^^i^'^'!}, ^ ^ 4 ^ ) - The restrictions (p{x,z) to CP^ \ {y = 0} and (p(y,z) to CP^ \ {x = 0} are defined in a similar way. The only transformations we consider are generic. This means that its Jacobian in nonzero, o[x, tj, z) We define the projective transformations / ^+,r+_,r : P'^ -^ CP^ by the formulas r_+(j; : y : z) = (( x) : y : z), r+_(.r : y : z) = (x : ( y) : z). and / _(./ : y : z) = (x : y : { z)). The projective transformations Si, .S2^ "^3 ' CP'^ ) CP^ are defined by the formulas ,SI(.T :y : Z) = (x : z : /y), S2(x : y : z) = (z : y : ./:), and s>^(.v : (/ : z) = {y : x : z). Finally we^ define the birational transformation liy h^' the formula hy(x : y : z) = (x^ : yz : xz) and its inverse transformation by hir^{x : y : z) = {xz : ./// : z"^). The transformation luj is called a hyperbolism, dm^ to Newton. 3 The set R = {id, r-+,r+_,r } is a group and has the composition of maps as the operation. It is generated by r_^ and r+_ and has the relations r^^ = r^_ = id and r^^ o r^^ = r+_ o / _+. The group R is isomorphic to Z2 x Z2. The s(^t of maps S = {id, Si, S2- Si o ^2, S2 o si, Si o S2 o 51} is a group with the operation as composition of maps. It is generated by Si and S2 and has the relations S | = .S2 ^<^ a^^d Si O S2 O S'l = S2 o Si o S2. The group 5 is isomorphic to the symmetric group 53. Note that S3 = Si o S2 o si. The set of integral powers of hy forms a free Abelian group H = {..., hy~^, hy~\ id. hy, hy-,...} that has the composition of maps as the operation. It is generated by hy and is isomorphic to Z. Let G = R*S*Hhe the group formed by the free product of R, S, and H. The and the set of relations of G is the generators of G is the set {r-+,r^_,si,S2,hy} union of the relations of R, S, and H. Let r_+ o Hy = / +_ o si / o Si .S'l o r_ + Si o r S i o / _,__ = = r _ + 0 S2 = = ^ = 1= S2 0 / __ S2 0 r + _ .S2 0 / _ + / +_ 0 ,S2 7^= < / 0 ,S2 hji / __|. 0 hy 0 r //// 0 r + _ r+_ 0 hy r _ _ 0 //// S2 0 hy hy 0 r_+ h(J~^ 0 ,S2 z=z be a s\steiii of relations. Definition 1.1.1 (Toric group). The factor group G/IZ with generators r_+, r+_, si, S2, hy is called the group of birational monomial transformations of CP^ or the toric group and is denoted T(CP^) . The relations 7t come from the algebraic properties of these transformations. The toric group T(CP^) is a subgroup of the Cremona group Cr(CP^ ). We can define the group T,{(CP'^) = {ip(x,y)\ip{x : y : z) e T(CP2)} which is isomorphic to T(CP^) and realizes the corresponding transformations in the affine chart C^ = (CP^ \ {z = 0}. The generators of T^(CP2) are r.+ {x,y) = {-x,y), r+_(x,y) = {x,-y), {x,y/x). Si{x,y) = {x/y,l/y), S2{x,y) = {l/x,y/x), and hy{x,y) = So it can be seen that the components of a transformation (p{x,y) are Laurent monomials, monomials of the form x'^^y^'^ with {uji,u}2) C Z^. The groups Tx(CP^) and Ty(CP^) are defined in a similar way and are isomorphic to T(CP^). The groups Ta;(CP^), Ty(CP^), and Tz{ P^) are called the groups of Laurent birational monomial transformations. There are four more elements of the group T(CP^) that are important for our application, so we will give them names. They are hx = s^ o hy o ss with inverse hx~^ = S2 o si o /ly o si o S2, ha = hxos2 with inverse ha~^ = ha, hb = s^o r^.- ohaos^ with inverse hb~^ = hb, and tr = si o hy~^ o s^o hy with inverse tr~^ = tr. They have the following formulas hx{x : y : z) {xz : y^ : yz), hx~^{x : y : z) = {xy : yz : z^), ha{x : y : z) = {xy : z^ : yz), hb{x : y : z) = {-z^ : xy : xz), and tr{x : y : z) {yz : xz : xy). The birational transformations ha, hb, and hx are hyperbolisms and tr is the standard triangle quadratic transformation. The transformation (p can be represented in coordinate form as ip{x:y:z) = {eix'^'y^'z^' : e2x''^y^H^' : x^'^y^^z^^) where ei,e2 C { 1,1} and the monomials x ''^y^^z^^,x^'^y^'^z^'^,x^^y^^z'^^ have no common factors. Since J{^) = ^^^/''^^'^"^ / 0, we have that d{x,y,z) ai X /3i 7i a 1-1-02-1-03-1 yl3i+l32+P3-l ^71-1-72+73-1 ^ Q g^^^ 0^2 02 72 3 A 73 /O. Since the determinate of the exponents is nonzero, the transformation cp is birational. Since the monomials have no common factors, one or two of a i , 0:25 0:3 and one or two of 3i, 02,03 and one or two of 71,72,73 is zero, also ai + /^i + 71 = 0:2 + /52 + 72 = <^3 + /^3 + 73- The degree of cp is ai -\- ^i -\- 71 and is denoted by degip. For any (p e Cr{([F^ ) , degip-'^ = degip. This is not always true for Cr(CP") with n > 3, [4]. 1.1.2 Space of Polynomials and Newton Polygons fifji -f- 3) Let RZZ be the linear space of the real homogeneous polynomials of degree n in the three variables x,y,z, which has dimension h 1. The set of monomials {x^^y^'^z^^\uji-\-uj2-\-uj3 = n} forms a basis in RZZ . Any polynomial f{x, y, z) C RZ/n can be represented in the form f{x,y,z)= Y^ uji+u!2+iji}3=n a^x'^'y'^^z'^^ u = {UJI,UJ2,UJ3). We now make the following definition. Definition 1.1.2 (Newton polygon). If f{Xi,X2,...,Xm) = wiH ^ \-uJm<n a^x'^' . . . X':;^, UJ = {uJi,...,UJm) is a polynomial of degree n, then the convex hull of the set {{uJi,.. , ujm) C R"^ la^; / 0} is called the Newton polygon of f{xi,X2,..., Xm)- U f{x, y.z) IS a homogeneous polynomial then we denote its Newton polygon by Nh{f). The substitution z = 1 in a homogeneous polynomial f{x,y, z) generates an isomorphism f{x,y,z) H^ f{x,y,l) from the linear space Ri/ onto the linear space R [./ , y] of real polynomials of degree < ii. If / C Rn[.^'. y] then we denote its Newton polygon l)y . (/) . If f{x,y,z) is a homogeneous polynomial then 1). .V(/) is understood to be the Newton polygon of f{x,y. The map pr ; R'^ ) R'^ is a projection defined by the formula pr(u;i, ^'2-^'3) = (^'i,.^'2) and the map li ; R^ ) R'^ is a lifting defined by the formula lin{uJi.uj2) = {u;i,uj2.n cji co'2)- If f{x,y,z) is a homogeneous polynomial of degree n, then {UJX,UJ2) V(/) = pr(A7?(/)) and Nh{f) = li (.Y(/)). The plane with coordinates is called the plane of Newton polygons and the space with coordinates (^'1, ^'2.u;3) is called the space of Newton polygons. A real Laurent polynomial is a finite linear combination of Laurent monomials with real coefficients. The Laurent polynomials form the ring R[a:. ?/..r~^. y~^]. .-\iiy Laurent polynomial f{x. y. 1) can be represented in the form f{x.y. where (/. j ) G Z^ and f^{x,y, f^{x.y, 1) = x^y^ f^{x.y. 1) 1) G R[a:,y] has no factors in .r and y. The polynomial 1). If ^ is a birational trans- 1) is said to be xy-free with respect to f{x.y. formation then (/^ o p)^ = (/ o (p)^. The Newton polygon A'(/'^) is obtained from the Newton polygon A^(/) by translation along the vector ( z, j) in the plane of Ncnvton polygons. Therefore the hat operation A : R[J:. y. .r"^, ?/~^] R[.r,/y] for a > given Laurent polynomial / = x'y^f^ defines the translation Tf : R'^ ) R'^ in the plane of Newton polygons. It is defined by the formula r/(cji.^2) = (^1 '-^2 j) such that r ; ( A ' ( / ) ) - .V(/^). A monomial transformation (p G r(CP'^) of degree d induces a homomorphism $^. : RZZ -^ RZZ , defined by the formula $ ^ ( / ) = f o ^"U Since ^^{oifi + r/2/2) = f'i*3>^(./i) + (i2^ip{f2) for all Oi,a2 G R and /1./2 G RZZn it is linear. .Also note that ker <I>. = 0. Let the transformation {x :y:z) = {e,x''^y'^z'^ : e2X^^y''^z^'^ : .r'^^//-^^) (1.3) have degree r/ and {x^'y^'^z^^\6i -\- 62-\- S3 = m} be a basis for RZZ^ . then the map (^. is defined in the bases of RZZ,, and RHm $^.(.r"''//"' -'c^'O = as ,.(^1 ^,62 ^(h x''y''z where ( ^1 = OL\ljJ\ \- OL2ijJ2 + (^1:5^3 (1.4) ^2 ^3 = = 0\UJ\ -f .^2^2 + '^3^'3 7X^1 + 72^2 + 73^3 and m 8x^-^2^ S3 nd. With the equation (1.3) we can define the map ,4 : T(CP2) -^ ,4(r(CP2)). The set .4(7(CP''^)) is a su])set of the group GZ/(3,R) of 3 x 3-matrices, given by the formula ai 02 ^3 A{p) = 01 02 03 \ 7l 72 ^;3 J We define the operation o by A{(p)oA{ip) = A{(pov^). This operation converts the set ,4(r(CP^)) of matrices into a group and converts the map .4 into a homomorphism of the groups. The matrix A{ip) is denoted by A^. For any p G 7"(CP^) the matrix A. is non-degenerate, since ker$(^ = 0. It can be shown that / 1 0 0 \ 4,V/ -Ti-r r- + , ^"i ' r+- 0 1 0 \ 0 0 1 / '^ 1 0 0 ^ ^-^.s. = '^ 0 0 1 ^ 4 I 2 0 1 -'^hy \^ 0 0 1 0 \ 1 0 0 \ 1 0 1 1 / V 0 1 0 I 1 0 0 I 0 For each monomial transformation (p G r(CP^) the system (1.4) defines a linear transformation A^ : R^ ^ R'^ in the space of Newton polygons. It is defined In- the formula A^p{uJi,LJ2,(jJ3) = {Si. 62. S3). The r(\striction p{x.y) of the transformation p{x : y : z) = {<pi{x : y : z) : p-2{-^' = 0} induces a map y : z) : p:>,{x : y : z)) oi degree d on the affine chart C = CP^ \{z (&^o(,,..y) : R,J.7',//] R rf[./ ,?/] defined by the formula > cIV(.,..,)(/(:r, y. 1)) = %{f){x. i,. z)\,=, = (/ o p-%r. 8 y. z)\,=y and induces an affine transformation B^^n : R^ -> R^ of the plane of Newton polygons defined by B^.,, = pr o .4^ o li / such that A'(/ o p-^) transformations Br_ + ,n{^l,UJ2) = -Pr+_,n ( ^ 1 , ^ 2 ) = (^1,^2), LJ2). = B-)^{N{f)). The affine - P . s i , n ( w i , ^ 2 ) = ( ^ 1 , n - UJiP.s>,n(Wl, ^ 2 ) = {nBhy,n{^1.^2) LUi - UJ2. UJ2), -UJ-2-^2), = {n-\-LUi ^/ly-1,71(^1,^2) = (^1+CJ2,^2) are area preserving. Since for any p G T(CP^) the transformation B^n is a composition of Pr_+,n. ^r+_,n, ^si,n, ^S2,n. Bhy^n,Bhy-\^^ the transformation B^.^ is also area preserving. Hence the areas of the Newton polygons A'(/) equal. 1.1.3 Mro Patchworking and A'(/ o p'^) are Consider the following polynomial fU'^y) = 5Z ^^-^"'^"^ ^ = (^1,^2) then we make the following definitions: Definition 1.1.3 (Quasihomogeneous). The polynomial f {x, y) i.s said to be quasihomogeneous if its Newton polygon is a segment. Definition 1.1.4 (Peripherally nondegenerate polynomial). A polynomial f is said to be peripherally nondegenerate if for every side F^ of its Newton polygon A'(/) tlie quasihomogeneous polynomial / ^ ' = and Definition 1.1.5 ( C o m p l e t e l y nondegenerate polynomial). .4 polynomial f ts sd/id to be completely nondegenerate if it is peripherally nondegenerate and does not hare any singular points in MP'^ \ [xyz = 0}. 9 X^WGT ^'^ ^'^"^V^' ^-^ nonsingular. Let us define the chart of a polynomial. First we give the following notation, for e,S e {+, - } we denote a quadrant by Qe,s = {{x, y) G R^ \ x > 0,Sy > 0}. Definition 1.1.6 (Chart of a peripherally nondegenerate p o l y n o m i a l ) . The chart of a peripherally nondegenerate polynomial f is the topological pair {N, v) such that the terms N and v are defined as follows: N: Consider the action R x R^ -^ R^ in the plane of Newton polygons. Let e,S e {+, }, then we denote the image of the Newton polygon N{f) r^s by Nss- So N is the orbit A/'++ U iV_+ U 7V+_ U N N{f) under this action. under the map polygon of the Newton v: This term is defined in two cases. First we consider the case where f is quasi-homogeneous. a segment and N = N^+UN-^UN-^-UN be a vector orthogonal to N{f) This means that N^s is {UJI,UJ2) is a union of segments. Let with relatively prime integer coordinates. We can see that the set R / is invariant under the transformation r'^\ o r^L We define the finite set v^s to be a subset of N^s such that the number of points of Ves is equal to the number of components of Rf fl Qes, for all e,S G {-\-, }. Also the union u++ U 'L'_+ U V+- U V is invariant under r^\ o r'^^_. So the second term is defined by v = v++ U f_+ U v+- U v The second case is when f is not quasi-homogeneous. is not empty and the boundary dN{f) segments by Ti,..., four steps. 1. Construct the charts of the quasi-homogeneous polynomials /^* corresponding to the sides F i , . . . , r . 2. For each e,S e { + , - } there is a homeomorphism h^s ' QeS -^ lntN{f) In this case lntN{f) consists of segments. We denote these Tn- The term v is a curve and is defined in the following which maps R/fl Qes C QeS onto a curve v^s C IntN^s- 10 3. The homeomorphism h = /i++U/i+_U/i_+U/i is chosen such that the pair coincides (5(IntA^++UlntA^+_UlntA^_+UlntA/'__),5(t'++Ui;+_U0_+U?)__)) with the union of the charts of the quasi-homogeneous polynomials / ^ ' , i = I,... ,n. The homeomorphism h has a natural correspondence between the = 0} near the coordinate axes in RP^ and of neard{lntN++UlntN+-UlntN-+UlntN ). Thus v = Cl(i)++ U branches of the curve Rf \{xyz the curve v++Uv+-Uv-+Uv 4. The curve v is the closure of the union of v^s i"^ N. v+- U i)_+ U {)__) = Cl{;++ U Cl{;+_ U Cl?)_+ U Cl{)__. / / we denote CXves as Ves, then we can express a chart by {N, v) (A^++, 'f++) U (7V+_, f+_) U (7V_+,i;_+)u(7V__,^;__). We denote a chart {N, v) of a polynomial f by V ( / ) . If V ( / ) is a chart of the polynomial / of degree n and p is one of the maps r_+, r+_, Si, S2, hy or a composition of them, then the chart V ( / o (p~^) can be easily regenerated by the chart V ( / ) . In particular, V ( / o r_+) = r_+(V(/)), V ( / o r+_) = r + - ( V ( / ) ) V ( / o si) = Bs,,n{N++, V++) U (r__ o Bs,,n o r+_)(iV+_, i;+_)U (r_+ o Bs n o r_+)(A^_+, v^+) U (r+_ o B,,, o r__)(A^__, t;__) V ( / o S2) = Bs,,n{N++, V++) U (r+_ o Bs,,n o r+_){N+.,v+.)U (r__ o Bs,,n o r_+)(A^_+, i;_+) U (r_+ o Bs,,n o r__)(A^__, i;__) ( / o hy) = B^j,-i, (Ar++,'u++) U (r+_ o B y-i,n o r+_)(A^+_,i;+_)U (r__ o B y-i, o r_+)(7V_+, i;_+) U (r_+ o B,,y-i, o r__)(A^__, i;__) V ( / o /12/-1) = Bhy,n{N^+. v+^) U (r+_ o Bhy,n o r+_)(iV+_, t'+_)U (r__ o BHy,n o r-+){N.+, 11 V-+) U (r_+ o B,,y, o r__)(A^__, c.__) As an example we show the regeneration of the map f{x, y, z) = -x -\-y -\- z hy hy~^, hy, tr, and ha. Figure 1.1. Figure 1.1: The curve f{x, y, z) = x -]- y -\- z and regenerations. The center of Figure 1.1 is the chart V(/). Starting in the upper left corner of Figure 1.1 and going around in the clockwise direction we get the charts: Y{f V(/ o hy), V(/ o tr~^), and V(/ o ha-^). Definition 1.1.7 (Viro's polynomials). The real polynomials fi{x, y, 1) satisfying the properties: 1. IntNif.) n IntiV(/,) = 0, Vi 7^ j fs{x, y. 1) ohy~^), 2. ^fN{f,) n N{f,) ^ 0, then / f ( "^</^) = /;(/.)nM/.), V/.,; :i the set N = U^^iN{fi) is convex are called Viro's polynomials. 12 It can be seen that there exists a unique polynomial / such that A'(/) = A^ with / ^ U ) = /^, V z - l , . . . , s . Definition 1.1.8 (Patchworking polynomials by z/). Let v : V ^ concave function such that 1. restrictions i'\N{f,) are linear functions, R^ be a 2. linearity of the restriction v\u to an open set U C N implies that the set U is contained m one of N{fi), 3. iy{N n Z2) c Z and f{x,y,i)= then let Yl ujeN{f)r\Z'^ ^^^'''y^' ft{x,y,l)= Yl M'^^"^^;'^^!/"^ the polynomials and we say that the polynomials ft are obtained by patchworking / i , . . . , /s by the function v. T h e o r e m 1.1.1. [ 1 9 ] Z / / i , . . . , / ^ are completely nondegenerate Uzro'.s' polynomials a/fid ft IS obtained by patchworking fi, . , fs by some nondegenerate concave function //. then there exists to > 0 such that for any t G (0, to] the polynomial ft is completely nondegenerate and its chart V(/i) is the union Uf^iY{fj) fg. = U^^i(A',, r,) = (Uj^jA^i, Ul^i'Ut) of charts of the polynomials f\, 13 CHAPTER II ENUMERATION AND RESTRICTION 2.1 Corollaries from Bezout and Harnack theorems In [16] Mro proved the classification of curves of degree 7. In particular he proved that there are only 14 distinct isotopy classes of AZ-curves of degree 7: (Jnani{^)) a + /3 = 14. 0</3<13 Let J be the odd branch of a curve of degree 7. One can easily check that there are six nonequivalent isotopy pairs (RP^ , J U RCi ) labeled as .4 - P and depicted in Figure 2.1. B D Figure 2.1: Isotopy pairs (RP'^ . J U RCi ) These isotopy pairs are listed from least to most complicated. Note that BezouUs theorem and Harnack's theorem easily imply that a nonsingular curve of degree 8 can only have one of the following schemes: 1) 2) 3) -1) ->) a) Ul{/5)> a < 22 a + ^<2l. ^ > 1 0 > 1 S>\ oni(^)ui(o)) rvUl(/i)Ul(-,)Ul((5)) ;i(l)Ul(l)Ul(l)Ul(l)) - U 1(;^ U 1{7))) V a + /5 + 7 < 20, 0>l, a 4 - / 3 + 7 + (5<19, / ? > 1 , 7 > 1 , <i) n -f /i + 7 < 20, ^i > L 7 > 1 -) (1{1(1{1 To prove this it is sufficient to apply Bezout"s theorem to the intersection of th(^ curv(\s of degree 8 with lines or conies. Note also that if a curv(> of degree // has onlv 14 double singular points then Bezout's theorem implies that perturbations of degree // of the double points are independent (See, for example, 2 [2]). These two notes immediately imply the following proposition. P r o p o s i t i o n 2.1.1. If the pair (1.2) is realizable by algebraic curves then, under smoothing its double points, one cannot obtain a curve of degree eight different from (i)-(VEach isotopy pair .4 - P in Figure 2.1 divides RP^ into 8 connected components. Noting the symmetries of each isotopy pair, we can see that the number of distinct regions for .4 is 2, B is 5, C is 5, Z> is 8, ^ is 5. and F is 5. By these facts, proposition : (2.1.1) admits 2 9^4 -h 5 9^4 ^ 5 9^4 ^ 8 9^4 + 5 9^^ + 5 9^4 = 30 ^^^ possible distributions of ovals. There is no need to list all these cases since most of them can be easily restricted. 2.2 Corollaries from the Gudkov-Rohklin congruence and its generalization W(^ want to start the prohibition of curves. To do this we need to introduce some terminology and theorems. It is known that if /(./ : y : z) I's a homogeneous polynomial of even degree then f{{ x) : { y) : { z)) = f{x : y : z). In this case one can define the sign of / in the complement RP^ \ Rf . The set RP^ \ R / is divided into two parts R P | and RP^ with the common boundary Rf . We chose RP^ to be the set which contains the non-oriented component and the sign of / such that IPl = {(./ : I, : z) G RP2 1/ > 0} and RP^ = {(:r : y : z) e W^ \f < 0}. The sets RP^ and RP^ are disjoint and each one consists of disjoint regions. Let rv b(^ an oval of the curve R / and D^ he its inner region. Let U he a neighborhood of (\ such that {Rf \a) nU = ^. The oval a is said to be positive if /(.r : tj : z) > 0 for (.7- : y : z) G f ^ 0 D^ and is said to be negative otherwise. We denote tli(^ number of positive ovals l)y p and the number of negative ovals by ti. Let g be a curv(^ of degree one and / a seventh degree curv(\ Then the curv( f(j = 0 is a curve of (>ven (l(>gr(>e, so w(^ can s])(H'if'v its ovals as (4ther positive^ or 15 iK^gative. The set RP^ \ Rf consists of regions homeomorphic to open disks, open disks with holes and one region homeomorphic to a Mobius band with holes. As above we let RPl he the set that contains the non-oriented region and choose the sign of / such that RPl = {{x :y : z) e RP^ \f > 0} and R P ! = [{X : y : z) G RP'^ \f < 0}. This is shown in Figure 2.2. Figure 2.2: Isotopy pairs (RP^ , J u RCi ) with signed regions Definition 2.2.1 (Generalized M-curves). A generalized M-curve is a curve C that satisfies the following three properties: 1. C is a reducible curve with no multiple factors. 2. each factor of C is an M-curve. 3. any two factors of C have the maximum number of points of intersection lying on one component of each factor of C. With these definitions we can now state the Generalized Gudkov-Rohklin Congruence. T h e o r e m 2.2.1 (Generalized Gudkov-Rohklin Congruence). If Rf is a generalized M-curve, then p n = m mod 8. all \Mi(^re for curves of tvp(\s .4 and C, tii 1; for D and E, m = 5; and for B. ni = 7. The value /// in the Generalized Gudkov-Rohklin congruence for a giv(Mi isotopy pair (1.2) is fix(>(l for all arrangements of the ovals. The following two theorems were also used in the prohibition of cur\(\s. 16 T h e o r e m 2.2.2. [15] If an eighth degree M-curve has the scheme {a Ul{3) U 1(-) U 1(6)), then l3,j,S = l mod 2. T h e o r e m 2.2.3. [16] If degree nonsingular (n H l{/3) U 1(7) U 1{S)) is the real scheme of an eighth ^ 0, then two of (5, 1, S are odd {M 2)-curve such that f^.j.S and tlie other is even. Using \'iro's classification of seventh degree AZ-curves, the Generalized GudkovRohklin congruence, BezouUs theorem, theorem 2.2.2, and theorem 2.2.3 we can restrict the possible arrangements of ovals for the isotopy pairs in Figure 2.1. To do this we first find an example of a curve and solve for m in the Gudkov-Rohklin congruence. Next we enumerate the possible arrangement of the ovals and prohibit cas(\s by using Bezout's theorem and theorem 2.2.2. Let us give an example for the isotopy pair .4. .All other cases are done in a similar way. In section 3.1 we construct the curve .43(12.1,2),(Figure 2.3). Figure 2.3: Constructed curve .43(12.1,2) The number of positive and negative ovals is p = 12 and ii = 3. This giv(>s us /// = p - II = 9 in the Gudkov-Rohklin congruence. To find the number of positive^ ovals we solve the system (2.1) for p. p- n = 9 p 4- n = 15 mod 8 mod 8 Th(> solution /; = 0 mod 4 iii(>aiis that the number of positive^ ovals for tli(^ is()toi)y pair .4 must be p G {0,4,8,12}. Next we prohibit certain arrangements of ovals for -4. One example is shown in Figure 2.4. To prove that this arrangement is prohibited we apply the small parameter method to the curve which gives us a curve of degree eight, Cg. Figure 2.4: A prohibited arrangement for .4. Let us pick a point in an oval from each collection a,.. .,e and run a conic C2 through those five points, as shown in Figure 2.5. Figure 2.5: C8UC2. The number of points of intersection for the curves is # ( ^ 8 H C2) = 18, but by Bezout's theorem #(6*8 fl C2) = 16. Another way to prohibit this curve is as follows. WYite the eighth degree curve in terms of Viro's coding scheme to get {a U l(/3) U. 1(7) II i{S) n 1(^)). By proposition (2.1.1) this is not one of the realizable curv(^s. Hence the curve in Figure 2.4 is nonrealizable. We would now like to stat(^ and prove a theorem on the prohibition of some curves. T h e o r e m 2.2.4. Let C be an M-curve of degree seven with isotopy type A.-,. .4(;, .47, .4,s, .4n, .4|2. or A\3. then p,j,S = 1 mod 2. 18 Proof. Let C he an AZ-curve of degree seven with isotopy type .45. .46. .47. .48. .4ii. .412, or .4i3. .Applying Harnack's method to C generates an eighth degree AZ-curve having the scheme { a m ( ^ ) U l { 7 ) n i { ( ^ ) ) . By theorem 2.2.2./3, 7, 5 = 1 mod 2. 2.3 Nonprohibited cases The cases that have not been prohibited are shown in Figures 2.6-2.37 and their corresponding tables are shown in Tables 2.1-2.5, giving the conditions, congruences, theorems applied, and number of solutions. D A. A. A:i{a.lS.y) Figure 2.6: Ai(a,l3):A;(a,l3.'r),A3(a,0,l), A. A, A- A8 Figure 2.7; .45(0, /3,7, S),Ae(Q,P- 7, <5), .h(a,d. 7,5)..48(ri. ,^,7. 5) A10 Figure 2.8; A.,{<\.P): .-lio(a.,i7) 19 ^11 A12 '13 M4 Figure 2.9: Au{a, (3,7, S),Au{a, /5,7. S), A,3{a, P, 7, S); Au{a. 0. 7) Table 2.1: Cases of arrangements for .4 # 1 2 3 4 5 6 / 8 9 10 11 12 13 14 T\-pe AI A2 .I3 .1.1 .45 -46 .47 .48 .19 -410 Fig 2.6 2.6 2.6 2.6 2.7 2.7 2.7 2.7 2.8 2.8 2.9 2.9 2.9 2.9 Conditions Q + ;9 = 15 Q - l - ^ + 7 = 15 Q + /3 + 7 = 15 ft + /3 + 7 = 15 Q + /3 + 7 + (5 = 15 a + 0 + j + 6 = 15 Q + ;8 + 7 + 5 = 15 n + /3 + 7 + d = 15 Q + /3 = 14 fl + ^ + 7 = 14 a + l3 + j + S - l i 0 + + 13 ^ + 6 = U /3>1 /3,7>1 ^,7>1 /3,7>1 /3,7,J>1 I3,1,S>1 0,1,6>l 0,1,6>l 0>1 0,1 > l 0,1,6 0,1,6 > 1 > 1 Congruence Q = 0 mod 4 Q = 0 mod 4 a = 0 mod 4 Q = 0 mod 4 Q = 0 mod 4 Q = 0 mod 4 a = 0 mod 4 Q = 0 mod 4 Q = 0 mod 3 Q = 0 mod 3 a = 0 mod 3 Q = 0 mod 3 n = 0 mod 3 a + 0 = 0 mod 4 Thm None None None None 2.2.4 2.2.4 2.2.4 2.2.4 None None 2.2.4 2.2.4 2.2.4 None # of solns 4 16 16 16 30 50 30 30 .3 18 14 14 14 24 4u .412 .4 13 .4i.i a + P + 'r + 5 = I a + P+ j = l ^,7,<5> 1 0,1> 1 B. B. B, Figure 2.10: Bi(n, l3).B2{n.l3),B:i{a. 6) 20 Figure 2.11: B,{a, (5. i),B,{a, 13, y),Be{a, p, 7), Bj{a. 0, i),Bs{a, p. -!).Bg{a, 3. -.) Figure 2.12: B,o{a, 0,i,S),..., Bi,{a. /S. 7, S) B16 B17 B18 Figure 2.13: Bie{a, 3);Bi7{a. 3. -f).Bi8{a. fS. 7) Figure 2.14: Big{(\. P)',B2o{a. 3.-,),B2i{a. 3.-). B22{n^3.-,) 21 Figure 2.15: ^23(0^, 0. I.S),..., B2s{a, (3, 7, S) B29 Figure 2.16: B29{a,p,j) 22 Table 2.2 : Cases of arrangements for B # 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 Type Bi B2 B3 BA Fig Conditions a + /3 = 15; a-1-/3 = 15; a + ^ = 15; Q + ;3 + 7 = 15; a + ;S + 7 = 15a + /3 + 7 = 15 a + /3-l-7 = 15 Q + /3 + 7 = 15 Q + ^ + 7 = 15 a + 0 + i + 6=15 a +0+i + 6-l5 0>i 0>1 0>1 0,1 0,1 0,1 0,1 0,1 >1 >1 >1 >1 >1 Congruence Q = 3 mod 4 Q = 3 mod 4 Q = 3 mod 4 a = 3 mod 4 a = 3 mod 4 Q = 3 mod 4 a = 3 mod 4 Q = 3 mod 4 a = 3 mod 4 a = 3 mod 4 a = 3 mod 4 >1 >1 >1 >1 0>1 0,1 > I 0,1 >1 0>1 0,1 >^ Q = 3 mod 4 a = 3 mod 4 a = 3 mod 4 Q = 3 mod 4 a + 0 = 3 mod 4 a + 0 = 3 mod 4 a + 0 = 3 mod 4 a = 2 mod 4 a = 2 mod 4 Q = 2 mod 4 Q = 2 mod 4 Q = 2 mod 4 Q = 2 mod 4 Q = 2 mod 4 a = 2 mod 4 Q = 2 mod 4 Q = 2 mod 4 Q + /3 = 3 mod 4 Thm None None None None None None None None None 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 None None None None None None None 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 None # of solns 4 4 4 21 21 21 21 12 12 66 66 35 35 64 66 15 21 21 3 21 21 21 66 66 66 66 36 36 21 2.10 2.10 2.10 2.11 2.11 2.11 2.11 2.11 2.11 2.12 2.12 2.12 2.12 2.12 2.12 2.13 2.13 2.13 2.14 2.14 2.14 2.14 2.15 2.15 2.15 2.15 2.15 2.15 2.16 Bs Be By Bs BQ 0,1 > I 0,1,6 0,1,6>1 0,1,6 0,1,6 0,1,6 0,1,6 >1 BIO fill Bl2 BIZ a + l3 + i + 6= 15 a + /3 + i + 6 = 15 a + 0 + i + 6 = 15 a +0+i + 6-l5 BlA Bl5 Bl6 Bn Bis B\9 B20 B21 B22 B23 B24 B25 B26 B27 B2S B29 Q + /3 = 15 a + 0 + i = 15 a + 0+i=l5 a +0 a +0+ ^U i^U a + 0 + 1 = 14 Q + /3 + 7 = 14 ; a +0+i + 6=U ; 0,1 > I /9,7>l 0,1,6 >i 0 + 0 + 1 + 6 = 14 ; 0 +0 +1+ 6=14 ; 0,1,6>l 0,1,6>i 0,1,6>l 0,1,6 0,1,6 >i >i a + 0 + i + 6 = 14 ; a + 0 + i + 6 = 14 ; a + 0 + 1 + 6 = 14 ; Q + ;S + 7 = 14 ; 0,1 > I Ci Figure 2.17: Ci{cy, P).C2{a, 0):C3{o. 3,-j).C,{n. 3.^): C,{n. 3,^.S) 23 Figure 2.18: C^{a,P,^),C,{a, p,-f),C8{a, P,^),C,{a, 3,-,) 10 11 Figure 2.19: Cio( , 0, j , S), Cn{a, 0,7, S) 12 13 14 15 16 Figure 2.20: Ci2{a, P);Ci3{a, 13,7), Cu{a, 13,7); Ci5(a, /?, 7, S), CiQ{a, l3, 7, S) 24 Table 2.3 : Cases of arrangements for C # 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 Type C'l (^-'2 Fig 2.17 2.17 2.17 2.17 2.17 2.18 2.18 2.18 2.18 2.19 2.19 2.20 2.20 2.20 2.20 2.20 Conditions a + ^ = 15; a + 0 = 15 Q + /3 + 7 = 15 a + ^ + 7 = 15 a + 0 + i + 6=15 0>i 0>i 0,1 0,1>1 0,1,6 0,1 0,1 0,1 >1 >1 >1 >1 >1 Congruence a = 0 mod 4 a = 0 mod 4 Q = 0 mod 4 Q = 0 mod 4 a = 0 mod 4 Q + ^ = 0 mod 4 a + 0 = 0 mod 4 a + 0 = 0 mod 4 Q + ^ = 0 mod 4 a + 0 = 0 mod 4 a + ^ = 0 mod 4 Q = 3 mod 4 a = 3 mod 4 >1 >1 >i Q = 3 mod 4 Q = 3 mod 4 a = 3 mod 4 Thm None None None None None None None None None None None None None None None None # of solns 4 4 32 16 84 24 24 24 24 112 112 3 18 18 61 35 Ca CA C5 Ce (-' 7 a+ l3 + -f = 15 Q + /3 + 7 = 15 a + /3 + 7 = 15 a + p + j=15 Cx Co do C'li Cv2 Cl3 Ci4 Cl5 Cl6 0,1 > I 0,1,6>1 0,1,6>1 0>1 0,1>1 0,1 0,1,6 0,1,6 a + 0 + i + 6 = 15 a + 0 + i + 6= 15 a + P = 14 a + 0 + i= a + 0 +1 14 =14 a + 0 + i + 6= 14 , a + 0 + i + 6 = 14 ; D D, D, D D, Figure 2.21: D,(ft, /J), L'2(<>, l3),D3(a, ,9). Di{a, /}), D^(a.:i) 25 Figure 2.22: De{a, 0,j),..., D,,{a, /5, 7) D21 D22 D23 D24 P>25(a, 0.7, S) D25 Figure 2.23: Du{a, (3,-f,S),..., D26 D27 D28 D29 Figure 2.24: L>26(a, 0^ 7), ^27(a. 0. l)^ P*28(n, 0, 7). ^29(n. .1 7) 26 D33 D34 D35 D35{a, (3,7, S) Figure 2.25: P>3o( , 0n.S),..., D36 D37 D38 D39 Figure 2.26: P>36(a, 0,7, S, e), P>37(a, 0, 7, S, e), D3s{a, /3, 7, S, e), D3g{a, (3, 7. S. e) D40 D41 D42 D43 3,^) Figure 2.27: D4o{a, P,j), D4i{a, (3,j), Di2{a, p,^),Di3{cu 27 D47 D48 D49 D,,{a, P, 7, S) Figure 2.28: Du{a, 0,j,S),..., D50 D51 D52 D53 Figure 2.29: Z:>5o(a, 0,7, S, s), D^i{a, P, 7, S, e), Dr,2{a, 13, 7, S, e), Dryi{a, [3, 7, S, e) D54 Figure 2.30; Dr,4{a,P) 28 Figure 2.31: D,,{a, (3, 7), D,e{a, p, 7). D,j{a. p, 7), D,s{a, (3,7), D,,{a. /5, 7) D65 D66 D67 D68 D^g{a, /3, 7, S) D69 Figure 2.32: Deo{a, p,j,S),..., 29 Table 2.4: Cases of arrangements for D # 60 61 62 (i3 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 91 95 96 97 98 Type Di D2 Fig 2.21 2.21 2.21 2.21 2.21 2.22 2.22 2.22 2.22 2.22 2.22 2.22 2.22 2.22 2.22 2.23 2.23 2.23 2.23 2.23 2.23 2.23 2.23 2.23 2.23 2.24 2.24 2.24 2.24 2.25 2.25 2.25 2.25 2.25 2.25 2.26 2.26 2.26 2.26 Conditions a + 0 = 15 a + 0 = 15 a + 0 =15 0>1 0>1 0>1 0>1 0>1 Congruence a = 1 mod 4 Q = 1 mod 4 a = 1 mod 4 Q = 1 mod 4 Q = 1 mod 4 a = 1 mod 4 Q = 1 mod 4 a = 1 mod 4 Q = 1 mod 4 Q = 1 mod 4 a = 1 mod 4 a = 1 mod 4 a = 1 mod 4 a = 1 mod 4 Q = 1 mod 4 Q = 1 mod 4 Q = 1 mod 4 Q = 1 mod 4 a = 1 mod 4 Q = 1 mod 4 a = 1 mod 4 a = 1 mod 4 Q = 1 mod 4 a = 1 mod 4 Q = 1 mod 4 a + 13 = 1 mod 4 a + 0 = 1 mod 4 a + 0 = 1 mod 4 a -j- ^ = 1 mod 4 Q -h /3 = 1 mod 4 a + 0 = 1 mod 4 Q -j- ;9 = 1 mod 4 a + 0 = 1 mod 4 a + 0 = 1 mod 4 n + /3 = 1 mod 4 a + 0 = 1 mod 4 a + 0 = 1 mod 4 a + 0 = 1 mod 4 a -|- ^ = 1 mod 4 Thm None None None None None None None None None None None None None None None 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 None None None None None None None None None None 2.2.3 2.2.3 2.2.3 2.2.3 # of solns 4 4 4 4 4 28 28 28 28 28 28 28 28 28 28 124 124 124 124 124 124 124 124 124 124 30 30 30 30 129 129 129 129 129 129 .348 348 348 348 Dz DA a-1-/3 = 15 a + / 3 = 15 Q-f-/3-)-7 = 15 Q + /3 + 7 = 15 Q - h ^ - f 7 = 15 Q - H ^ + 7 = 15 D5 De Dy Ds DQ Dio Dii a +0 + ^ = 15 a 4-,3 4 - 7 = 15 a - 1 - ^ - ^ 7 = 15 Q - h / 3 - H 7 = 15 Q - h ^ - | - 7 = 15 a + 0 + ^ = 15 a + /3 + i + 6 = 15 a + 0 + i + 6= 15 a + 0 + i + 6 = 15 a + 0 + i + 6 = 15 a + 0 + i + 6= 15 a + P + l + 6 = 15 a + 0 + i + 6= 15 a + 0 + i + 6= 15 a + 0 + i + 6= 15 a + 0 + i + 6 = 15 a + 0 + i = 15 a + 0 + i = 15 a + 0 + 1 = 15 a + 0 + a + 0 + i + i=15 6=15 Dv2 Diz DiA ^15 ^16 0 0 0 0 0 0 0 0 0 0 7>1 7>1 7>1 7>1 7>1 7>1 7>1 7>1 7>1 7>1 0,1 ,6 > 1 0,1 ,6> 1 0,1 ,6> 1 0,1 ,6> 1 0,1 ,6 > 1 0,1 6 > 1 0,1 6 > 1 0,1 6 > 1 0,1 6 > 1 0,1 6 > 1 Dn Dis Dig D20 D21 D22 D23 D24 D25 D26 D27 D28 D29 D30 D31 D32 D33 D3A D35 D36 D37 D3S D39 0, 7 > 1 0, 7 > 1 0, 7 > 1 0, 7 > 1 0,1 6> 1 0,1 6> 1 0,1 6> 1 0,1 6> 1 0,1 6> 1 0,1 6 > 1 0,1,6 e > 1 0,1,6 ^ > 1 0,1,6 > 1 0,1,6 > 1 a + 0 + i + 6= 15 a + 0 + i + 6= 15 a + 0 + i + 6= 15 a + 0 + i + 6=15 a + 0 + i + 6= 15 a + 0 + 1 + 6 + = 15 a + 0 + i + 6 + = 15 a + 0 + i + 6 + = 15 a + 0 + i + 6+ =l5 30 # !)!) Type ^40 DAI DA2 DA3 DAA DA5 DA6 DAI DAH DA9 Fig 2.27 2.27 2.27 2.27 2.28 2.28 2.28 2.28 2.28 2.28 2.29 2.29 2.29 2.29 2.30 2.31 2.31 2.31 2.31 2.31 2.32 2.32 2.32 2.32 2.32 2.32 2.32 2.32 2.32 2.32 Conditions a + /3 -f 7 = 15 a + 13 +-f = 15 a + 0 + ^ = 15 a + 0 + ^ = 15 a + 0 + 'y + 6 = 15 a + 0 + i + 6 = 15 a + 0 + l + 6 = 15 a + 0 + i + 6 = 15 a + 0 + i + 6 = 15 a + 0 + i + 6 = 15 a + 0 + i + 6 + E = 15 a + 0 + i + 6 + = 15 a + 0 + i + 6 + = 15 a + p + i + 6 + = 15 a + P = 14 a + 0 + 1 = 14 a + 0 + i = 14 a + 0 + i = 14 a + 0 + 1 = 14 a + 0 + i = 14 a + 0 + i + 6 = 14 a + 0 + i + 6 = 14 a + 0 + i + 6 = 14 a + 0 + i + 6 = 14 a + 0 + i + 6 = 14 a + 0 + i + 6 = 14 a + 0 + i + 6 = 14 a + 0 + i + 6 = 14 a + 0 + i + 6 = 14 a + 0 + i + 6 = 14 0,1> 1 Congruence a + P = 1 mod 4 a + 0 = 1 mod 4 a + P = I mod 4 a + 0 = 1 mod 4 a + 0 = 1 mod 4 a + 0 = 1 mod 4 a -I- /3 = 1 m o d 4 a + 0 = 1 mod 4 a + P = 1 mod 4 a + 0 = 1 mod 4 a + 0 = 1 mod 4 a + P = 1 mod 4 1 a + P = 1 mod 4 a + P = 1 mod 4 1 Q = 0 mod 4 a = 0 mod 4 a = 0 mod 4 a = 0 mod 4 Q = 0 mod 4 >1 >1 Q = 0 mod 4 a = 0 mod 4 a = 0 mod 4 >1 >1 >1 a = 0 mod 4 a = 0 mod 4 a = 0 mod 4 Q = 0 mod 4 Q = 0 mod 4 >1 >1 >1 Q = 0 mod 4 Q = 0 mod 4 Q = 0 mod 4 Thm None None None None None None None None None None 2.2.3 2.2.3 2.2.3 2.2.3 None None None .None None None 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 2.2.3 # of solns 30 30 30 30 129 129 129 129 129 129 348 348 348 348 4 28 28 28 28 28 124 124 124 124 124 124 124 124 124 124 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 0,1 > 1 0,1> 0,1> 0,1,6 0,1,6 0,1,6 0,1,6 0,1,6 0,1,6 0,1,6, > 0,1,6, >1 0,1,6, > 0,1,6, >1 0> 1 1 > 1 > 1 > 1 > 1 > 1 > 1 1 D50 D51 D52 D53 DsA D55 D56 D57 D58 D59 /9,7>1 0,1>1 0,1>1 0,1>1 0,1 0,1,6 0,1,6>1 0,1,6 0,1,6 0,1,6 0,1,6>1 0,1,6>1 0,1,6 0,1,6 0,1,6 Deo Dei De2 De3 DQA Des Dee De7 Des De9 Figure 2.33: Ei{(\, 0), E2{(y, 0)',E3{(y, 0-l), E4{c\. p.j); E:,{n.0.i,S) 31 10 11 Figure 2.34: Ee{a, 0,j),..., En{a, (3,7) 15 16 17 Figure 2.35: ^12(0;, 0, J,S),..., Eu{a, /?, 7, S) 18 19 20 21 22 Figure 2.36: Eis{(y, 0,-f,S,e),Eig{a, l3,j,S,e)', E2i){(u 3): E2i{(\. 3.-^), E>>{n, 3,j) 32 23 24 Figure 2.37: E23{a, (3,7, S), E2,{a, p, 7, S) Table 2.5: Cases of arrangments for E # 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 Type El E2 E3 EA Fig 2.33 2.33 2.33 2.33 2.33 2.34 2.34 2.34 2.34 2.34 2.,34 2.35 2.35 2.35 2.35 2.35 2.35 2.36 2.36 2.36 2.36 2.36 2.37 2.37 Conditions a - f ^ = 15 a - F / 9 = 15 0>1 0>1 0,1 0,1 0,1,6>1 0,1 >1 >1 >1 Congruence Q = 2 mod 4 Q = 2 mod 4 a = 2 mod 4 Q = 2 mod 4 Q = 2 mod 4 a + P = 2 mod 4 a + P = 2 mod 4 a + P = 2 mod 4 a + P = 2 mod 4 a + P = 2 mod 4 a + P = 2 mod 4 a + P = 2 mod 4 a -1- ^ = 2 mod 4 a + P = 2 mod 4 Q -1- ,9 = 2 mod 4 a + P = 2 mod 4 >1 1 a -t- /3 = 2 mod 4 Q -1- /9 = 2 mod 4 a + ^3 = 2 mod 4 a = 1 mod 4 Q = 1 mod 4 Q = 1 mod 4 Q = 1 mod 4 a = 1 mod 4 Thm None None None None None None None None None None None None None None None None None None None None None None None None # of solns 4 4 12 24 56 32 32 32 32 32 32 112 112 112 112 112 112 360 360 4 24 2-1 56 100 a + p + ^=\5 a + p + 'y = 15 a + P + i + 6= 15 a + P + i = 15 a + p + 'y=15 E5 Ee E7 Es Eg Eio Ell E12 Ei3 ElA Ei5 0,1 > 1 0,1 > 1 0,1 > 1 0,1 > 1 0,1 > 1 0,1,6> 0,1,6>1 0,1,6>1 0,1,6 > 1 1 Q - h / 3 - h 7 = 15 Q + ^ + 7 = 15 a + P+ i a + P+ i =15 =15 a + P + i + 6 = 15 a + P+ i + a + P+ i + a + P+ i + a + p + i + a + p + i + 6=15 6=15 6=15 6=15 6=15 Eie El7 E18 Ei9 E20 E21 E22 E23 E2A 0,1,5>1 0,1,6 0,1,6, > 0,1,6, >1 /3>1 0,1 > 1 0,1 > 1 0,1,6 0,1,6 > 1 >1 a + P + i + 6 + = 15 a + P+ i + 6+ =15 a + P = 14 a + p + i a + P+ i =14 =14 a + P + i + 6= 14 a + P+ i + 6=14 33 The Table 2.6 gives the total number of cases for the arrangement of ovals for each is()t()i)y pair and the total number of cases for all the isotopy pairs. The entry for the case F is empty since we have not constructed a curve of this type and therefore cannot restrict enough arrangements to be helpful. Table 2.6: Total number of cases Case Number of cases A B C D E F Total 279 921 595 6457 1893 No Data 10145 34 CHAPTER III CONSTRUCTION WV used two different methods to construct seventh degree .U-curves with the maximum number of points of intersection with a line. The first method of constructing curves is \4ro's Method. We used two methods to generate the \4ro polynomials used in the gluing. The other method used was Harnack's method. We start with \ 4 r o s method. 3.1 Construction using \4ro's Method We are now ready to start the construction of our curves. The first step is to constnict curves of fourth degree having a planar point at ( 1 : 0 : 0). Some of the.se curves must have four points of intersection with a line and the others must have three points of intersection with a line. Let us start by constructing these curves. To construct these curves we start with the curve x = y^ and apply the transformation hy^^ to it. The charts of the curve and transformations are shown in Figure 3.1. hy-1 Figure 3.1: The curve x = y^ and the transformation hy ^ Next we need the charts shown in Figures 3.2. 35 Figure 3.2: Third-degree curves with three points of intersection These are the charts of third degree curves having the maximum number of points of iiit(us(Hti()ii with two lines and the maximum number of ovals. One was constructed by Viro [17] and the other three by Korchagin [6]. We glue the charts from Figure 3.2, after the appropriate transformation, with the chart from Figure 3.1-2 to get fourth degree curves with the maximum number of ovals, a planar point at (1 : 0 : 0). and four points of intersection with a line. The results are shown in Figure 3.3. Figure 3.3: Fourth-degree curves, maximum number of ovals, planar point at (1 : 0 : 0), 4 points of intersection N(>xt w(^ applied different transformations to the charts in Figure 3.2 and again glued th(^m with the chart in Figure 3.1-2 to get the charts of fourth degree curves having a planar point at ( 1 : 0 : 0 ) , three points of intersection with a line, and the maximum number of ovals (see Figure 3.4). 36 Figure 3.4: Fourth-degree curves, maximum number of ovals, planar point at (1 : 0 : 0), 3 points of intersection After the appropriate transformation and shift of the axis of the charts in Figure 3.3-1, we generate the charts in Figure 3.5. Figure 3.5: Fourth degree curve, planar point at (1 : 0 : 0), four points of intersection .Appling the appropriate transformation to the chart in Figure 3.4-2 we get the charts in Figure 3.6. 1 Figure 3.6: Fourth degree curve, planar point at ( 1 : 0 : 0), three points of iiitt^iscHtioii Next we transform the charts in Figure 3.5 by r+_o/ia and the charts in Figure 3.6 i)v hb to get the charts in Figures 3.7 and 3.8. Figure 3.7: Figure 3.5 charts transformed by r+_ o ha Figure 3.8: Figure 3.6 charts transformed by hb Now we glue the charts in Figures 3.7 and 3.8 to get the charts in Figure 3.9. 11 12 21 22 31 32 41 42 Figure 3.9: Glued charts from Figures 3.7 and 3.8 Finally we need curves of degrees four that have the maximum number of i)oints of intersection with a line, have no singularities and do not pass throught tlu^ points (0 ; 0 : 1), (0 : 1 : 0). (1 : 0 : 0). The charts of these curv(\s are shown in Figure 3.10. 38 o 6 7 8 Figure 3.10: Fourth-degree curves with four points of intersection We are now ready to apply Viro's method to construct seventh degree .U-curves having the maximum number of points of intersection with a line. First, we take a chart from Figure 3.9 and note how the curve intersects the edges. Next, we take the charts from Figure 3.10 and multiply them by y^ which separates the upper and lower halves of the charts. We then pick the charts that when the upper half is glued to the upper edge and the lower half is glued to the lower edge of the chart from Figure 3.9 we get new ovals. We also pick charts from Figure 3.2 such that when multiplied ])>' .r^ the left and right halv(>s match up with the left and right edges of the chart from Figure 3.9 to produce new ovals. Gluing all allowable combinations of these charts from Figures 3.10 and 3.2 with the charts from (3.9) produces .^IZ-curves of degree seven having the maximum number of points of intersection with a line. This method does not guarantee that all the intersections will occur on one component. In fact, gluing (3.9.31) with (3.10.6) and (3.2.1) produces a seventh degree TlZ-curve having seven points of interscH-tion, but some points are on the odd branch and some are on an oval. As an example, if we take the charts (3.9.11). (3.10.1), and (3.10.2) and glue them as specified ab()v(> we generate the i\Z-ciirv(^ of type .43(12,1,2). The curv(\s generat(Ml bv this m(4hod are shown in Figures 3.11-3.18. 39 11.3.1 4(10,2,8) 11.3.2 Bg(11,3,1) 11.4.1 4(10,1,4) 11.4.2 Bg(11,3,1) Figure 3.11: 7*'^ degree M-curve by gluing (3.9.11) with (3.10.1-4) and (3.2.1-2) 12.1.3 62(11,4) 12.1.4 A.,(12,3) 12.2.3 Bg(11,1,3) 12.2.4 A3(12,1,2) 12.3.3 2(10,5) 12.3.4 B2(11.4) 12.4.3 4(10,3,2) 12.4.4 Bg(11.2,2) Figure 3.12: 7^^ degree M-curve by gluing (3.9.12) with (3.10.1-4) and (3.2.3-4) 40 21.1.1 63(11,1,3) 21.1.2 A3(12,1,2) 21.2.1 62(11,4) 21.2.2 A^(12,3) 21.3.1 4(10,3,2) 21.3.2 66(11,2,2) 21.4.1 2(10,5) 21.4.2 62(11,4) Figure 3.13: 7^" degree ^Z-curve by gluing (3.9.21) with (3.10.1-4) and (3.2.1-2) 22.1.3 Bg(11,1,3) 22.1.4 A3(12,1,2) 22.2.3 63(11.2,2) 22.2.4 A3(12,1,2) 22.3.3 4(10,1,4) 22.3.4 Bg(11,3,1) 22.4.3 4(10,2,3) 22.4.4 Bg(11,3,1) Figure 3.14: 7^^ degree 7\Z-curve by gluing (3.9.22) with (3.10.1-4) and (3.2.3-4) 41 31.5.1 627(6,1,2,5) 31.5.2 A^2(7.1.1.5) 31.6.1 Nonmaximal Intersection with the Odd Branch 31.6.2 Nonmaximal Intersection with the Odd Branch 31.7.1 24(5,1,2,6) 31.7.2 625(6,1,6,1) 31.8.1 Nonmaximal Intersection with the Odd Branch 31.8.2 Nonmaximal Intersection with the Odd Branch Figure 3.15: 7*^ degree M-curve by gluing (3.9.31) with (3.10.5-8) and (3.2.1-2) 32.5.3 821(6,1,7) 32.5.4 10(7.1.6) 32.6.3 Nonmaximal Intersection with the Odd Branch 32.6.4 Nonmaximal Intersection with the Odd Branch 32.7.3 22(5,1,8) 32.7.4 821(6,1,7) 32.8.3 Nonmaximal Intersection with the Odd Branch 32.8.4 Nonmaximal Intersection with the Odd Branch Figure 3.16: 7*^ degree M-curve by gluing (3.9.32) with (3.10.5-8) and (3.2.3-4) 42 41.5.1 Nonmaximal Intersection with the Odd Branch 41.5.2 Nonmaximal Intersection with the Odd Branch 41.6.1 621(6,1,7) 41.6.2 Ai 0(7.1.6) 41.7.1 Nonmaximal Intersection with the Odd Branch 41.7.2 Nonmaximal Intersection with the Odd Branch 41.8.1 22(5.1,8) 41.8.2 621(6,1,7) Figure 3.17: 7'^ degree M-curve by gluing (3.9.41) with (3.10.5-8) and (3.2.1-2) 42.5.3 Nonmaximal Intersection with the Odd Branch 42.5.4 Nonmaximal Intersection with the Odd Branch 42.6.3 627(6.1,2,5) 42.6.4 Ai2(7.1,1.5) 42.7.3 Nonmaximal Intersection with the Odd eranch 42.7.4 Nonmaximal Intersection with the Odd eranch 42.8.3 24(5,1,2.6) 42.8.4 625(6.1.6.1) Figure 3.18: 7*^ degree M-curve by gluing (3.9.42) with (3.10.5-8) and (3.2.3-4) After checking the constructions, we found that the following curves have points of intersection other than on the odd branch: 3.15.6.1, 3.15.6.2, 3.15.8.1, 3.15.8.2, 3.16.6.3, 3.16.6.4, 3.16.8.3, 3.16.8.4, 3.17.5.1, 3.17.5.2, 3.17.7.1, 3.17.7.2, 3.18.5.3. 3.18.5.4, 3.18.7.3, and 3.18.7.4. 43 3.2 Construction of Korchagin's curves The next constructions are performed by taking sixth-degree curves with a Jio singular point at (0 : 0 : 1) having six distinct points of intersection with a line and unioning a line tangent to the singularity. This gives us a seventh degree curve with an A'21 singular point at (0 : 0 : 1) and having seven distinct points of intersection with the line. If we then smooth this singular point in the proper way, we will obtain .\Z-curves having the maximum number of points of intersection with a line. \AV will first construct the sixth-degree curves described above, which were constructed by Korchagin [5]. There are two types of these curves, meaning that each t>'pe intersects the line in two different ways. The first type looks like a "hook" where it int(usects the line and the second looks like a "wave." We will construct the "hook" first. We start with the conic C2 such that it is tangent to 2:0 = 0 at the point ( 0 : 0 : 1 ) and has two distinct points of intersection with X2 = 0. This is shown in Figure 3.19. Figure 3.19: Conic C2 Using the small parameter method we construct C2 = C2 3- tLiL2 = 0, where Li and L> ar(> distinct lines that pass through ( 0 : 0 : 1 ) . Figure 3.20. 44 Figure 3.20: Conic C2 with construction curves Again using the small parameter method we construct C4 = 02(72 + ^6/4 = 0, where OA = ./(J + x\. This is shown in Figure 3.21. Figure 3.21: Curve C4 with construction curves Let ^6 be two distinct parabolas tangent to each other and tangent to ./Q = 0 at the point ( 0 : 0 : 1 ) and a line transversal to the parabolas also passing through ( 0 : 0 : 1 ) . Then we construct the curve C5 = C^XQ + td^ = 0. This produces two curvets, C'l and C | , having a singularity at (0 : 0 : 1), intersecting the line X2 = 0 at fi\-e distinct points and the maximum number of ovals. Figure 3.22. 45 Figure 3.22: Curves Ci and Cl Let jio be three distinct parabolas tangent to each other and tangent to XQ = 0 at ( 0 : 0 : 1 ) . Finally we construct the curves C'^^ = C^XQ + fjio = 0. This produces four curves Cl'\cl''^.Cf;\ and Cg'^ shown in Figure 3.23. Figure 3.23: Curves Cl^\Cl''\C'^:\ and C,6 2,2 To construct the second type of curve we again start with a conic C> tangent to .ro = 0 at (0 : 0 : 1) and intersecting X2 = 0 at two distinct points, Figure ;i24. 46 Figure 3.24: Conic C2 Then we construct C^ = C2XQX2 + tLiL2L3L4 = 0, where the lines L, = 0 are distinct, pass through ( 0 : 0 : 1 ) , and are situated as shown in Figure 3.25. Figure 3.25: Curve C4 with construction curves Next we construct C5 = C4X0 -f- tde. This produces the curves Q and C'^^ having a singularity at ( 0 : 0 : 1), intersecting the line X2 = 0 at five distinct points and the maximum number of ovals (Figure 3.26). 4^ Figure 3.26: Curves C^ and C? The final step is to construct the curves Cg'^ = C^XQ + tjio = 0. Again this produces four curves C^''^,Cl''^,Cl'\ and CQ'^, shown in Figures 3.27. Xo=0 .2,2 ^1,1 Al,2 A2,l Figure 3.27: Curves C^^\C,'\Ct^ and C, The curves Q ^ and C'Q^ i,j G {1,2} are sixth degree curv(\s that have a Jio singular point at (0 : 0 : 1) and intersect the line .r2 = 0 at six distinct points. 48 3.2.1 Construction using Korchagin's curves We are now ready to construct new seventh degree M-curves having the maximum number of points of intersection with a line. We construct the seventh degree curves Cj = CQ-'XQ = 0 and Cj'-' = Cg'-^xo = 0. These curves have an A'21 type singular point at (0 : 0 : 1) and intersect the line 3:2 = 0 at seven distinct points. The curves look (^xactly like Figures 3.23 and 3.27, except now the line XQ = 0 is part of the curve. Before we construct these curves we want to show how to smooth the A'21 singular point. The A^2i singular point and three of the results of smoothing are shown in Figure 3.28 with corresponding tables. Table 3.1. These were found by Viro [17]. B Figure 3.28: A'21 singularity with the smoothings. Table 3.1: Fi)^ure 3.28 A, B, and C fV 8 1 0 4 5 0 8 0 1 4 3 2 0 7 2 4 1 4 4 0 5 0 1 8 a 7 1 3 5 0 7 0 49 Let us start by defining some curves needed to smooth the singularity. Each of these curves has a singular point at (0 : 0 : 1) and the parabolic branches are tangent to .1Q = 0, except OQ which does not have a singular point but is still tangent to XQ = 0. The curves aj, = ay l3x^ = 0 consists of one parabolic branch. If a/3 < 0 the parabola opens up and is denoted by aj. If a/3 > 0 it opens down, denoted al. The curves a\ = {(\ 1 y 3ix){a2y 02^^) = 0 consists of one parabolic branch and a line transversal to it. there are two such curves. The next curves are a\ = {aiy (3ix'^){a2y 32X^) = 0 which consists of two parabolic branches. There are four curves of this type. The curves dl = {aiy (3ix){a2y 02x'^){oi3y 03x'^) = 0 consists of two parabolic branches and a line transversal to them. Again there are four curves of this type. The curves J\Q = {aiy - 0ix'^){o-2y - 02x'^){a3y - 33x'^) = 0, z G { 1 , . . . , 8 } , consists of three parabolic branches. The curves zjg = {aiy (3ix){a2y (32^'^){<^3!J 33x'^){(^4y 34X^) = 0, i C { 1 , . . . , 8 } , consists of three parabolic branches and a transversal line. Finally we have the curves x^i = y{aiy - pix'^){a2y - 02-^''^){(^3y - 33X^), 8} = 0, which consists of three parabolic branches and a line tangent to I G {1 them. The singular points that correspond to these curves are shown in Figure 3.29 A,0 A. A, D, Jio '15 X 21 Figure 3.29: Singularities .4(), .4i, .4,{, Do, Jio, ^i5- and A21 )0 To smooth the singular point A'21 we use the small parameter method and start with the curve Q21 = ^'i = 0, where ii G { 1 , . . . , 8 } . First we generate the curve ^'z\5 = ^x-21 + ^-^15 = 0? where t is a small parameter and Z G { 1 , . . . , 8 } - From these 2 curvc^s we generate the curves Cjlo^'" = Cl'^i + tj\l = 0, 23 G { 1 , . . . . 8}. Continuing w(^ generate the curves Q^^2^3M = ql^2^3 ^ ^^u ^ Q, Cl\'''''''' ^ qU2r,^A^.^e = Q m ^ 3 ^ 4 ^ 5 _^ ^^Z6 ^ Q, a n d finally C'^Q'''''''''''' = Q^2^3^4 _^ ^^^5 ^ Q Q. = Q/2^3^4^5^6 ^ ^^J^r ^ Where A G { 1 , . . . , 4 } , is G { 1 , . . . , 4 } , Z G {1, 2}, and 27 G {1,2}. 4 g Not all of the curves generated above have the desired properties. Picking those that have the maximum number of ovals which have not been constructed before and relabeling the curves as C^Q, this gives us eighth degree curves that have no singular points and in a neighborhood about ( 0 : 0 : 1 ) look like Figure 3.28.A-C, with the number of ovals shown in Table 3.1. To construct seventh degree M-curves with the maximum number of points of intersection with a line from the curves Cj\ C]'-', and C^Q, we use Viro's method again. First we construct the charts of C\'^ and C]'\ shown in Figures 3.30 and 3.31. 51 I, Figure 3.30: Charts of C7 J )2 Figure 3.31: Charts of Cf^ Next we construct the charts for the smoothings in Figure 3.28. They are shown in Figure 3.32. Figure 3.32: Charts of smoothings. Since the smoothings are of degree eight we must apply a transformation to them to get degree seven curves. The smoothings consist of four parabolic branches tangent to each other at the point ( 0 : 0 : 1 ) . They have the equation Cs = {y - ax^){y - (3x^){y _ ^:,2^^y _ ^^2) ^ y-^^^^j ^ Q^ ^^^^^ a > /3 > 7 >(5. We apply the transformation {x, y) ^{x,y^^^^^^^c^rveCl^, tx') to Cg. Setting t = S we get the seventh + f{x,y^tx') =0 = {y-{a-t)x'){y-{P-t)x^){y-{j-t)x^)y The charts of C^ ^g are shown in Figure 3.33. Figure 3.33: Transformed charts of smoothings. Applying the appropriate transformations to C^^Q we glue the them with the charts from Figures 3.30 and 3.31 to generate the seventh degree .U-curves having the maximum number of points of intersection with a line, shown in Figures 3.34 to 3.41. 54 Figure 3.34: Constructed from 3.30 and 3.33 Figure 3.35: Constructed from 3.30 and 3.33 JO Figure 3.36: Constructed from 3.30 and 3.33 Figure 3.37: Constructed from 3.30 and 3.33 56 Figure 3.38: Constructed from 3.31 and 3.33 Figure 3.39: Constructed from 3.31 and 3.33 0/ Figure 3.40: Constructed from 3.31 and 3.33 Figure 3.41: Constructed from 3.31 and 3.33 58 3.3 Harnack's Method: i\Z-curves of degree 6 We applied Harnack's method to the 7\Z-curves of degree six having the maximum numb(n- of points of intersection with a line (Figure 3.42) to generate JZ-curves of degree seven having the maximum number of points of intersection with a line. Figure 3.42: Some iU-curves of degree six. The curves labeled 5, 8, 9, 16, 17, 18, 30, and 31 in Figure 3.42 generated new .U-curves having seven points of intersection with a line. The curves below were constructed from the curves in Figure 3.42 that lunv the "comb" isotopy tvpe and are shown in Figures 3.4.3-3.59. )1) ^ - ' ; - ' : ooooo oooooo Figure 3.43: Seventh-degree M-curve Aio(ll, 1,2) constructed from Figure 3.42-1. OOOOOO oooooo Figure 3.44: Seventh-degree AZ-curve .4i(12,3) constructed from Figure 3.42-2. 60 Figure 3.45: Seventh-degree M-curve ^io(7,5,2) constructed from Figure 3.42-3. oooo oooo Figure 3.46; Seventh-degree M-curve ^i(8, 7) constructed from Figure 3.42-4. 61 Figure 3.47: Seventh-degree M-curve Aio{7,3,A) constructed from Figure 3.42-5. Figure 3 .48: Seventh-degree M-curve Aro{7.1.6) constructed from Figure 3.42-6, 62 Figure 3.49: Seventh-degree M-curve ^io(ll, 1,2) constructed from Figure 3.42- ' ' > ' > '' ooo Figure 3.50: Seventh-degree M-curve Aio(3,5,6) constructed from Figure 3.42-8. 63 Figure 3.51: Seventh-degree M-curve ^io(3,2,9) constructed from Figure 3.42-9. Figure 3.52: Seventh-degree M-curve y4io(ll, 1,2) constructed from Figure 3.42-14. 64 ;pj ;p) ;p) ;b)' ;o'i ooo oooo Figure 3.53: Seventh-degree M-curve Aio(7,5,2) constructed from Figure 3.42-15. 1 ; ' ^^ .^ , , ; OOO oooo Figure 3.54: Seventh-degree M-curve Aio(7,4,3) constructed from Figure 3.42-16. 65 p,i ;p) (O); p) ;p) ;p)i :'Pi ;'pi V Figure 3.55: Seventh-degree M-curve ^io(3,8,3) constructed from Figure 3.42-17. ; o i (Qi ; o i ; o i ,' I / / ;p,i ;p; ;p,i ;pj \ \ ,,'' / /,'"~- \ /' /' \ / \; 1' j \ 1 / \; \ '--, 1/ ,'-\^ \ / ,'"\ \ ' /"\ \ Oii/ ^ ^ ^-' ^-' o) ooo Figure 3.56: Seventh-degree M-curve Aio(3,1,10) constructed from Figure 3.42-18. 66 Figure 3.57: Seventh-degree M-curve Aio(7,1,6) constructed from Figure 3.42-19. oooooo OOOOO Figure 3.58: Seventh-degree M-curve ^14(11,1,2) constructed from Figure 3.42-30. 67 ooo oooo Figure 3.59: Seventh-degree M-curve ^14(7,1,6) constructed from Figure 3.42-31. Table 3.2: Table of constructed curves from Figure (3.42) Figure # 3.42-1 3.42-2 3.42-3 3.42-4 3.42-5 3.42-6 3.42-7 3.42-8 3.42-9 3.42-14 3.42-15 3.42-16 3.42-17 3.42-18 3.42-19 3.42-30 3.42-31 Curve generated Aio(ll,l,2) Ai(12,3) Aio(7,5,2) Ai(8,7) Aio(7,3,4) Aio(7,l,6) Aio(ll,l,2) Aio(3,5,6) Aio(3,2,9) Aio(ll,l,2) Aio(7,5,2) Aio(7,4,3) Aio(3,8,3) Aio(3,l,10) Aio(7,l,6) Ai4(ll,l,2) Ai4(7,l,6) Figure 3.43 3.44 3.45 3.46 3.47 3.48 3.49 3.50 3.51 3.52 3.53 3.54 3.55 3.56 3.57 3.58 3.59 68 CHAPTER I\' CONCLUSION We would like to make the following conjectures. C o n j e c t u r e 4.0.1. If the curve is of type Bu{a,p,^,S), then 7 = 0 mod 2. C o n j e c t u r e 4.0.2. If the curve is of type B23{a, ^, 7. S), then S = 0 mod 2. C o n j e c t u r e 4.0.3. The isotopy pair F is nonrealizable for M-curves. We would like to give the motivation for conjecture (4.0.3). Recall that in our constructions we never constructed a curve with the isotopy pair F. In the classification of sixth degree iU-curves having the maximum number of points of intersection with a line on one component, there are four ways for the line to intersection a component, shown in Figure (4.1) and listed from least to most complicated. "comb" "snake" "snail" "camel" Figure 4.1: Isotopy pairs for degree six iU-curves It was proved by Shustin that the "Camel" is not realizable for ilZ-curves of degree six. For our problem the isotopy pair F is the most complicated, therefore we suspect that it is also not realizable. Finallv we would now like to give the tables showing the curves construct(Hl. 69 Table 4.1: Constructed curves of isotopy type .4 Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Type Al AI A2 A2 A3 AA AA AA AA AA AA AY a 8 12 8 12 12 4 4 8 8 8 12 4 4 8 8 8 12 3 3 3 3 7 7 7 7 11 7 11 7 7 11 0 7 3 6 2 1 4 10 2 4 6 2 3 9 1 3 5 1 1 2 5 8 1 3 4 5 1 5 1 1 1 1 1 6 Figure 3.41 3.12 1 1 2 7 1 5 3 1 1 1 1 1 1 1 1 10 9 6 3 6 4 3 2 2 1 1 1 6 2 1 1 5 7 1 5 3 1 1 3.40 3.40 3.11 3.41 3.41 3.41 3.41 3.41 3.41 3.40 3.40 3.40 3.40 3.40 3.40 3.56 3.51 3.50 3.55 3.16 3.47 3.54 3.41 3.41 3.40 3.40 3.15 3.59 3.58 Aj AT AT AT AT Aio Aio Aio Aio Aio Aio Aio Aio Aio All All Al2 Ai4 Al4 70 Table 4.2: Constructed curves of isotopy type B Number 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 Type B2 B2 BA BA a 7 11 7 11 11 11 3 3 7 7 7 11 11 11 3 3 7 7 7 11 6 6 10 6 10 6 6 0 8 4 6 2 2 3 5 11 3 5 7 3 1 2 3 9 1 3 5 1 1 5 1 5 1 1 1 7 6 Figure 3.37 3.12 2 2 2 1 7 1 5 3 1 1 3 2 2 2 2 2 2 2 7 3 3 1 1 6 2 2 2 1 5 7 1 5 3 1 1 3.36 3.36 3.12 3.11 3.37 3.37 3.37 3.37 3.37 3.37 3.11 3.11 3.36 3.36 3.36 3.36 3.36 3.36 3.16 3.37 3.37 3.36 3.36 3.15 3.15 Be Be B7 B7 B7 ST ST B7 Bs Bs Bn Bn Bn Bn Bn Bn B21 B21 B21 B23 B23 B25 B27 71 Table 4.3: Constructed curves of isotopy type C Number 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 Type Cs Cs Cs Cs Cs Cs CA CA a 12 8 8 4 8 4 12 8 11 7 12 8 8 4 8 4 12 8 11 7 0 1 5 3 7 1 1 2 6 1 5 1 5 3 7 1 1 3 7 1 5 7 1 1 3 3 5 9 1 1 1 1 2 2 4 4 6 10 6 1 1 1 1 1 1 Figure 3.38 3.38 3.38 3.38 3.38 3.38 3.38 3.38 Cie Cie Cs Cz C3 C3 C3 C3 C2 C2 CiA CiA 1 1 3.38 3.38 3.39 3.39 3.39 3.39 3.39 3.39 3.39 3.39 2 2 3.39 3.39 '2 Table 4.4: Constructed curves of isotopy type D Number 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 Type DT DT a 13 9 9 5 9 5 13 9 12 8 13 9 9 5 9 5 13 9 12 8 0 1 5 3 7 1 1 2 6 1 5 1 5 3 7 1 1 2 6 1 5 7 1 1 3 3 5 9 6 Figure 3.34 3.34 3.34 3.34 3.34 3.34 3.34 3.34 Dj D-r D7 D7 D3 Dz D57 D57 1 1 1 1 3 3 5 9 3.34 3.34 3.35 3.35 3.35 3.35 3.35 3.35 3.35 3.35 Ds Dg Ds Ds Ds Ds DA DA D5S D58 1 1 3.35 3.35 Table 4.5: Constructed curves of isotopy type E Number 99 100 101 102 103 Type E2 EA EA a 10 10 10 5 5 0 5 1 2 1 1 7 6 Figure 3.12 4 3 8 2 6 3.11 3.11 3.16 3.15 E22 E2A 73 BIBLIOGRAPHY '^' ^N'^SSST^' ^er \Z''^ li^TAZ^ ^^' ^'P'^'^y of real algebraic curves. Izvestiva I SsSst1yV'21:l\S '^' N m ^ L ? 9 Q ^ 1 Q 7 T i ? ^ ' ^ o " ? n ^ ^ ' ^ ' ' ^ ^^'""'^^ ^^^'^' ''''''' ^ '''''' ^ ^^^^^^^^'^ ^ ^^^^^'e^' Uspekhi Mat. S r l ^ N o 4 1.79 '^' ?0718^6)^ 189^-%'^9^^^ ^- ^"^^' ^^^^^'^ '^^^'i- ^^ ^^^ i^^ -^^^^^i^- s ^- - - - Vol ^ i^^^^ '^^'' ^^'''^'' a/^e6mz5c/zen i^wr^en Math. Ann., m plane and space Cambridge Univ. '^' Pr^e^r^l^27^'' ^''^'^'''^'' transformation [.j] Korchagin A. B., On the reduction of singularities and the classification of nonsingular ajjine curves of degree 6, Deposited at \TNITI N o . 1107-B86, (1986), [6] Korchagin A. B., Smoothing of 6-Fold Singular Points and Constructions of 9'^ Degree M-Curves, Amer. Math. Soc. Transl., 173:2(1996), 141-155. [7] Korchagin A. B., The 1st Part of HilberVs 16th Problem: History and Results Visiting Scholar's Lectures, Texas Tech Univ. Math Series 19(1997), 84-139. [8] Korchagin A. B., On Toric Transformations, (To appear), (2001). [9] Mishachev N. M., The complex orientations of the plane M-curves of odd degree Func. Analysis AppL, 9:4 (1975), 77-78(Russian). [10] Rokhlin V. A., Congruence modulo 16 in Hilbert sixteenth problem, Func Analysis AppL, 6:4 (1974), 58-64(Russian). [11] Rokhlin X A., The complex orientations of the real algebraic e/arves. Func Analysis AppL, 6:4 (1974), 71-75(Russian). [12] Rokhlin \ . A., The complex topoloigical characteristics of the real algebraic curves, Uspekhi Math. Nauk, 33:5 (1978), (Russian). English transl. in Russian Math. Surveys Vol 33(1979). [13] Viio O. Ya., Constructing 78 (Russian). M-surfaces. Func. Analysis .-VppL 13:3 (1979). 77- [14] \'iro O. Ya., Constructing multicomponent real algebraic surfaces, Dolk. AN SSSR 248:2 (1979), 1038-1041(Russian); English transl. in Soviet Math. Dolk., 20 (1979), 991-995. [15] Yiro O. \ a . , The curves of degree 7, the curves of deqree 8, and Raqsdale e(mjecture, Dolk. AN SSSR 254:6 (1980), 1305-1310 (Russian); English transl. in Soviet Math. Dolk. 22(1980), 566-570. 4 [16] Viro O. Ya., Plane real curves of degrees 7 an 8: the new prohibitions. Izvestiya AN SSSR, ser Mat., 47:5(1983). 853-863 (Russian); English transl. in Math USSR Isvestiya 23:2(1984), 409-422. [17] Viro O. Ya., Real algebraic varieties with prescribed topological properties, D. Sc. Thesis, Leningrad State Univ., 1983. (Russian) [18] Viro O. Ya., Gluing of the plane real algebraic curves and constructions of curves of degrees 6 and 7, Lect. Notes Math. 1060 (1984), 185-299. [19] Viro O. Ya., Patchworking real algebraic varieties, Preprint Univ.(Sweden), Dept. Math, report 20 (1994), l-50(English)?. Uppsala iO PERMISSION TO COPY In presentmg this thesis in partial fulfillment of the requirements for a master's degree at Texas Tech University or Texas Tech University Health Sciences Center, I agree that the Library and my major department shall make it freely available for research purposes. 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