An effective description of the roots of bivariates mod pk and the related Igusa’s local zeta function
Pages 135 - 144
Abstract
Finding roots of a bivariate polynomial f(x1, x2), over a prime field, 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 roots to . Such roots may be numerous and behave unpredictably, i.e., they may or may not lift from to .
We give the first algorithm to describe the roots of a bivariate polynomial over 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 p2k 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.
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Index Terms
- An effective description of the roots of bivariates mod pk and the related Igusa’s local zeta function
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