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lecture-4

# lecture-4 - Polynomials Given Nn a monomial in n variables...

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Polynomials I Given α N n , a monomial in n variables is a function m α : R n R defined as m α ( x ) := x α 1 1 x α 2 2 · · · x α n n . I The degree of a monomial is defined as deg m α := n i =1 α i . I A polynomial in n variables is a function p : R n R defined as a finite linear combination of monomials: p := X α ∈A c α m α = X α ∈A c α x α where A ⊂ N n is a finite set and c α R α ∈ A . I The set of polynomials in n variables { x 1 , . . . , x n } will be denoted R [ x 1 , . . . , x n ] or, more compactly, R [ x ] . I The degree of a polynomial f is defined as deg f := max α ∈A ,c α 6 =0 deg m α . I θ R [ x ] will denote the zero polynomial, i.e. θ ( x ) = 0 x . 18/235

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Multipoly Toolbox I Multipoly is a Matlab toolbox for the creation and manipulation of polynomials of one or more variables. I Example: pvar x1 x2 p = 2*x1^4 + 2*x1^3*x2 - x1^2*x2^2 + 5*x2^4 I A scalar polynomial of T terms and V variables is stored as a T × 1 vector of coefficients, a T × V matrix of natural numbers, and a V × 1 array of variable names. p .coef = 2 2 - 1 5 , p .deg = 4 0 3 1 2 2 0 4 , p .var = x1 x2 19/235
Vector Representation I If p is a polynomial of degree d in n variables then there exists a coefficient vector c R l w such that p = c T w where w ( x ) := 1 , x 1 , x 2 , . . . , x n , x 2 1 , x 1 x 2 , . . . , x 2 n , . . . , x d n T l w denotes the length of w . It is easy to verify l w = ( n + d d ) . I Example: p = 2*x1^4 + 2*x1^3*x2 - x1^2*x2^2 + 5*x2^4; x = [x1;x2]; w = monomials(x,0:4); c = poly2basis(p,w); p - c’*w [c w] 20/235

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Gram Matrix Representation I If p is a polynomial of degree 2 d in n variables then there exists a Q ∈ S l z × l z such that p = z T Qz where z := 1 , x 1 , x 2 , . . . , x n , x 2 1 , x 1 x 2 , . . . , x 2 n , . . . , x d n T The dimension of z is l z = ( n + d d ) . I Equating coefficients of p and z T Qz yields linear equality constraints on the entries of Q I Define q := vec ( Q ) and l w := ( n +2 d 2 d ) . I There exists A R l w × l 2 z and c R l w such that p = z T Qz (i.e. p ( x ) - z ( x ) T Qz ( x ) θ ) is equivalent to Aq = c . I There are h := l z ( l z +1) 2 - l w linearly independent homogeneous solutions { N i } h i =1 each of which satisfies z T N i z = θ ( x ) . I Summary : All solutions to p = z T Qz can be expressed as the sum of a particular solution and a homogeneous solution. The set of homogeneous solutions depends on n and d while the particular solution depends on p . 21/235
Gram Matrix Example p = 2*x1^4 + 2*x1^3*x2 - x1^2*x2^2 + 5*x2^4; [z,c,A,w] = gramconstraint(p); p-c’*w Q = full(reshape(A\c,[3 3])); p-z’*Q*z % Q is a particular solution in vectorized form % Each column of N is a homogenous solution in vectorized form. [z,Q,N] = gramsol(p); Q = full(reshape(Q,[3 3])); N = full(reshape(N,[3 3])); p-z’*Q*z z’*N*z z = x 2 1 x 1 x 2 x 2 2 , Q = 2 1 - 0 . 5 1 0 0 - 0 . 5 0 5 , N = 0 0 - 0 . 5 0 1 0 - 0 . 5 0 0 22/235

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Positive Semidefinite Polynomials I p R [ x ] is positive semi-definite (PSD) if p ( x ) 0 x . The set of PSD polynomials in n variables { x 1 , . . . , x n } will be denoted P [ x 1 , . . . , x n ] or P [ x ] .
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lecture-4 - Polynomials Given Nn a monomial in n variables...

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