An effective description of the roots of bivariates mod p(k) and the related Igusa's local zeta function

Finding roots of a bivariate polynomial f (x(1),x(2)), over a prime field F-p, is a fundamental question with a long history and several practical algorithms are now known. Effective algorithms for describing the roots modulo pk,k >= 2, for any general bivariate polynomial, were unknown until the present paper. The main obstruction is lifting the singular F-p roots to Z/p(k)Z. Such roots may be numerous and behave unpredictably, i.e., they may or may not lift from Z/p(j)Z to Z/p(j+1)Z. We give the first algorithm to describe the roots of a bivariate polynomial over Z/p(k)Z in a practical way. Notably, when the degree of the polynomial is constant, our algorithm runs in deterministic time which is polynomial in p + k. This is a significant improvement over brute force, which would require exploring p(2k) possible values. Our method also gives the first efficient algorithms for the following problems (which were also open): (1) efficiently representing all the (possibly infinitely-many) p-adic roots, and (2) computing the underlying Igusa's local zeta function. We also obtain a new, effective method to prove the rationality of this zeta function.