Newton–Hensel Interpolation Lifting

Abstract

The main result of this paper is a new version of Newton-Hensel lifting that relates to interpolation questions. It allows one to lift polynomials in Z[x] from information modulo a prime number p = 2 to a power p k for any k, and its originality is that it is a mixed version that not only lifts the coefficients of the polynomial but also its exponents. We show that this result corresponds exactly to a Newton-Hensel lifting of a system of 2t generalized equations in 2t unknowns in the ring of p-adic integers Z p . Finally, we apply our results to sparse polynomial interpolation in Z[x].

Interpolation of polynomials given by straight-line programs

“…Avendaño, Krick and Pacetti [2] obtain a heuristic algorithm for the interpolation of polynomials in Z[X ], with a complexity polynomial in h, log(d),h, p, whereh is a bound on the heights of A(y i ), for some suitable evaluation points y i ∈ Z, and where p is a ''lucky'' prime greater than τ . However, the probabilistic aspects are not…”

“…To state it, we use the O˜( ) notation, so as to omit logarithmic factors: 2 + n 4 τ 4 log(d) 3 ) operations in S and a similar number of bit operations.…”

A new deterministic algorithm for sparse multivariate polynomial interpolation

“…Avendaño, Krick and Pacetti [4] studied sparse polynomial interpolation over Z [x] and C[x]. They developed a new version of Newton-Hensel lifting specifically for interpolation questions.…”

“…They developed a new version of Newton-Hensel lifting specifically for interpolation questions. For t-sparse polynomial f ∈ C[x] represented by straight line programs of length L, Avendaño et al [4] found an algorithm which interpolates such f in time O(t 4 L). Also, they studied interpolation over (Z/pZ)[x].…”

“…Also, they studied interpolation over (Z/pZ)[x]. Avendaño et al [4] found an algorithm which interpolates t-sparse polynomials in (Z/pZ) [x] in time O((t 2 +p) log p). But this requires values of the polynomial at special points {ρ i } 0≤i<2t , here ρ ∈ Z is a primitive root modulo p. Their new version of Newton-Hensel lifting allows the interpolation over Z[x] from interpolation over…”