6.885 Algebra and Computation
November 21, 2005
Lecture 20
Lecturer: Madhu Sudan
1
Scribe: Eitan Reich
Today
Hilberts Nullstellensatz
2
Generic/Random Linear Transformations
Given an ideal in K [x1 , ., xn ], our goal is to nd a basis for this ideal that
6.885 Problem Set: Due November 14, 2005
Instructions
If you are taking the course for credit, you must turn in this problem set.
You should do this in pairs. If you have trouble nding a partner let me know and Ill help
you nd one.
While a pair may ask
6.885 Algebra and Computation
November 23, 2005
Lecture 21
Lecturer: Madhu Sudan
1
Scribe: Michael Manapat
Quantied Statements
Let : K n cfw_T, F be a Boolean function dened as follows:
(x1 ) = x1 x2 x3 Qw xw ,
Q a quantier, such that
fi (x0 , . . . , xw
November 28, 2005
6.885 Algebra and Computation
Lecture 22
Lecturer: Madhu Sudan
Scribe: Krzysztof Onak
Today we present various algebraic models of computation, and discover a few lower bounds.
1
Algebraic models of computation
1.1
Considered problems
Le
6.885 Algebra and Computation
October 31, 2005
Lecture 14
Lecturer: Madhu Sudan
Scribe: Sophie Rapoport
Today, we give a brief review of factorization of polymnomials, then begin our study of primality
testing by giving some simple but inecient algorithms
6.885 Algebra and Computation
November 14, 2005
Lecture 18
Lecturer: Madhu Sudan
1
Scribe: Alexey Spiridonov
Today
We cover Grbner basis recognition, generation, and the resulting algorithm for ideal membership.
o
We wont produce any complexity estimate
6.885 Algebra and Computation
November 7, 2005
Lecture 16
Lecturer: Madhu Sudan
1
Scribe: Sergey Yekhanin
Today
Decoding of Reed-Solomon (RS) codes.
Decoding of Chinese Remainder (CR) codes.
2
Error-correcting codes
The theory of error-correcting codes
6.885 Algebra and Computation
November 9, 2005
Lecture 17
Lecturer: Madhu Sudan
Scribe: Paul Valiant
The problem which we consider today is the following: given m polynomials in n variables over a
eld K ,
f1 , ., fm K [x1 , ., xn ],
def
does there exist a