Algebra and Computation by Madhu Sudan

By Madhu Sudan

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1 Motivating Discussion Let f be a multivariate polynomial over the integers, that is f(x1 ; x2; : : :; xn) 2 Zx1; : : :; xn]. We want to check if f is irreducible. Clearly, this can be done using an exhaustive search. Hilbert asked if this can be done more e ciently. His suggestion was to substitute the variables, that is, let x1 = a1; x2 = a2 ; : : :; xn = an, where ai 2 Z, and check if f(a1 ; a2; : : :; an) is a prime. If f is reducible, say f(x1 ; x2; : : :; xn) = g(x1; x2; : : :; xn) h(x1; x2; : : :; xn); then \usually" g(a1 ; a2; : : :; an) 6= 1 and h(a1 ; a2; : : :; an) 6= 1 and f(a1 ; a2; : : :; an) is composite.

If either u or v is 0, then by hypothesis kz k2 kak2. If both u = 6 0 and v = 6 0 Case (i) : u > v We rst observe that (a + b) (a + b) b b ) 2a b a a kz k22 = (u2a a + v2 b b + 2uva b) (u2a a + v2 b b uva a) (u(u v)a a + v2 b b) (u(u v)a a) kak22 Case (i) : u v We now observe that (a + b) (a + b) ) 2a b kz k22 a a b b (u2 a a + v(v u)b b) (u2 a a) kak22 For the case n > 2, we employ the LLL basis reduction to nd a small vector. The LLL basis reduction will be discussed in the next lecture. 1 Introduction In the last lecture, we saw how nding a short vector in a lattice plays an important part in a polynomial time algorithm for factoring polynomials with rational coe cients.

Multiples of a polynomial p. The set of bivariate polynomials in x; y with monomials xiyj such that i + j d. Ideals are important because we can de ne the quotient R=I, where p q (mod I) if p q 2 I, and operations on R=I are well de ned. The product of two ideals I; J is the set of all nite sums of products of elements of I and J: n X I J = f aibi : ai 2 I; bi 2 J g: i=1 For example, if, for some p 2 R, Ip is the ideal fpq : q 2 Rg (denoted (p)) then I 2 = I:I = (p2). d. f; g; 2 R and ideal I in R, we say that pseudo gcd(f; g) = 1 (mod I) to imply that there exist a; binR such that af + bg = 1 (mod I).

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