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Lecture Notes

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Algebra and Computation

15 years 3 months ago

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These are the lecture notes by Madhu Sudan on the course Algebra and Computation, offered in MIT in 1999.

Madhu Sudan

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Added	26 Jan 2009
Updated	26 Jan 2009
Year	1999
Authors	Madhu Sudan

The Table of Contents

Lecture 1: Overview; Preliminaries.

Lecture 2: Polynomials; The factorization and GCD problems.

Lecture 3: Polynomials and error-correcting codes. The Euclidean algorithm for GCD. Applications. Resultant.

Lecture 4: Properties of the resultant. Applications. Towards factorization of polynomials: Finding square roots modulo a prime.

Lecture 5: Factorization of univariate polynomials over finite fields.

Lecture 6: Berlekamp's deterministic algorithm for factoring univariate polynomials. Existence, density and properties of irreducible polynomials.

Lecture 7: Factoring bivariate polynomials. Hensel's lifting.

Lecture 8: Factoring bivariate polynomials (contd.). Digression: Applications to error-correction algorithms.

Lecture 9: Irreducibility testing; Black-box factoring of multivariate polynomials.

Lecture 10: Factoring polynomials over the rationals; Reduction to basis reduction.

Lecture 11: LLL's Basis reduction algorithm.

Lecture 12: (2nd Phase of course) Ideals and Varieties. Division in Ideals; Groebner bases.

Lecture 13: Construction and uniqueness of Groebner bases. Solution to the Ideal Membership problem.

Lecture 14: Upper bound on the degrees for ideal generation. Complexity lower bound.

Lecture 15: Varieties. Emptiness of a variety. Elimination. Hilbert's Nullstellensatz.

Lecture 16: Strong form of Hilbert's Nullstellensatz. Quantifier Elimination.

Lecture 17: Quantifier elimination (contd.); Bezout's Thm.

Lecture 18: Bezout's Thm. and some applications.

Lecture 19: Algebraic models of computation; Ben-Or Cleve result.

Lecture 20: Blum-Shub-Smale model of computation (contd.) Undecidability of Mandelbrot Set.

Lecture 21: Algebaic settings for the P=NP question

Lecture 22: An Arthur-Merlin proof for the Hilbert's Nullstellensatz.

Lecture 23: Non-uniform lower bounds; Linear independence; algebraic independence; Strassen's degree bound.

Lecture 24: Ben-Or's lower bounds based on no. of connected components; Survey of other methods: Volume; Euler characteristic and Betti numbers.

Lecture 25: Mulmuley's Algebraic PRAM without bit operations. Lower bounds for LP and Max Flow.

Lecture 26: Summary. What we didn't cover ..

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