ACM Communications in Computer Algebra, Vol. 49, No. 3, Issue 193, September 2015 Using Sparse Interpolation to Solve Multivariate Diophantine Equations Michael Monagan and Baris Tuncer Department of Mathematics Simon Fraser University Burnaby, BC, CANADA, V5A 1S6 mmonagan@cecm.sfu.ca,ytuncer@sfu.ca 1 Introduction Suppose that we seek to factor a multivariate polynomial a ∈ R = Z[x 1 ,...,x n ] and a = fg with f,g in R. The multivariate Hensel lifting algorithm (MHL) developed by Wang [1] uses a prime number p and an ideal I = 〈x 2 - α 2 ,...,x n - α n 〉 of Z p [x 1 ,...,x n ] where α 2 ,α 3 ,...,α n ∈ Z p is an evaluation point chosen by the algorithm. For a given polynomial h ∈ R, let us use the notation h j := h(x 1 ,...,x j ,x j +1 = α j +1 ,...,x n = α n ) mod p so that a 1 = a(x 1 ,α 2 ,...,α n ) mod p. The input to MHL is a,I,f 1 ,g 1 and p such that a 1 = f 1 g 1 and gcd(f 1 ,g 1 ) = 1 in Z p [x 1 ]. If we denote by d j the total degree of a j with respect to the variables x 2 ,...,x j and I j = 〈x 2 - α 2 ,...,x j - α j 〉 with j ≤ n, Wang’s incremental design of MHL lifts the factorization a 1 = f 1 g 1 step by step to a j = f j g j ∈ Z p [x 1 ,...,x j ]/I d j +1 j . If p is sufficiently large, we recover the factorization of a over Z. In this process for each j with j ≤ n Wang’s algorithm must solve several instances of the multivariate Diophantine problem (MDP) that typically takes almost all of the factorization time. Let u, w, c ∈ Z p [x 1 ,...,x j ] in which u and w are monic polynomials with respect to the variable x 1 with j  1 and I j = 〈x 2 - α 2 ,...,x j - α j 〉 be an ideal of Z p [x 1 ,...,x j ] with α i ∈ Z p . The MDP consists of finding multivariate polynomials σ, τ ∈ Z p [x 1 ,...,x j ] that satisfy σu + τw = c mod I d j +1 j with deg x 1 (σ) < deg x 1 (w), in which d j is the maximal total degree of σ and τ with respect to the variables x 2 ,...,x j and it is given that 1. GCD(u, w) | c and 2. GCD (u mod I j ,w mod I j ) = 1 in Z p [x 1 ]. It can be shown that the solution (σ, τ ) exists and is unique provided the second condition is satisfied and that the solution is independent of the choice of the ideal I j . For j = 1 the MDP is in Z p [x 1 ] and can be solved with the extended Euclidean algorithm. For the multivariate case where j> 1, Wang uses the same approach as in Hensel lifting, that is, an ideal-adic expansion approach. In general, if α k = 0 for j ≤ k, then an 〈x k - α k 〉-adic expansion of the solution is expensive to compute. Since even a sparse solution turns out to be dense in an 〈x k - α k 〉-adic expansion, the computation of error becomes expensive. In this poster we present the results of experiments using various approaches of sparse interpolation to compute σ. Then, to compute τ , we use τ =(c - σu)/w for the case j> 1. 94