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Article Dans Une Revue Journal of Symbolic Computation Année : 2021

Computing the multilinear factors of lacunary polynomials without heights

Bruno Grenet

Résumé

We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in ℓ n where ℓ is the lacunary size of the input polynomial and n its number of variables, that is in particular polynomial in the logarithm of its degree. We also provide a randomized algorithm for the same problem of complexity polynomial in ℓ and n. Over other fields of characteristic zero and finite fields of large characteristic, our algorithms compute the multilinear factors having at least three monomials of multivariate polynomials. Lower bounds are provided to explain the limitations of our algorithm. As a by-product, we also design polynomial-time deterministic polynomial identity tests for families of polynomials which were not known to admit any. Our results are based on so-called Gap Theorem which reduce high-degree factorization to repeated low-degree factorizations. While previous algorithms used Gap Theorems expressed in terms of the heights of the coefficients, our Gap Theorems only depend on the exponents of the polynomials. This makes our algorithms more elementary and general, and faster in most cases.

Dates et versions

hal-00936318 , version 1 (24-01-2014)

Identifiants

Citer

Arkadev Chattopadhyay, Bruno Grenet, Pascal Koiran, Natacha Portier, Yann Strozecki. Computing the multilinear factors of lacunary polynomials without heights. Journal of Symbolic Computation, 2021, 104, pp.183-206. ⟨10.1016/j.jsc.2020.04.013⟩. ⟨hal-00936318⟩
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