We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in ${\ell }^n$ 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.

我们提出了一种确定性算法，该算法可计算数量字段上多元Lanary多项式的多线性因子。它的复杂度是多项式${\ell }^{ñ}$其中ℓ是输入多项式的底数大小，n是变量的数量，特别是多项式的对数。我们还提供了复杂多项式在同一问题的随机算法ℓ和ñ。

在特征零的其他字段和大特征的有限字段上，我们的算法将计算至少具有三个多项式多项式单项式的多线性因子。提供下界以解释我们算法的局限性。作为副产品，我们还为未知的多项式族设计多项式时间确定性多项式恒等式检验。

我们的结果基于所谓的间隙定理，该定理将高级分解分解为重复的低级分解。尽管先前的算法使用间隙定理来表示系数的高度，但我们的间隙定理仅取决于多项式的指数。这使我们的算法更加基础和通用，并且在大多数情况下更快。