Computing low-degree factors of lacunary polynomials: a Newton-Puiseux approach

We present a new algorithm for the computation of the irreducible factors of degree at most $d$, with multiplicity, of multivariate lacunary polynomials over fields of characteristic zero. The algorithm reduces this computation to the computation of irreducible factors of degree at most $d$ of univariate lacunary polynomials and to the factorization of low-degree multivariate polynomials. The reduction runs in time polynomial in the size of the input polynomial and in $d$. As a result, we obtain a new polynomial-time algorithm for the computation of low-degree factors, with multiplicity, of multivariate lacunary polynomials over number fields, but our method also gives partial results for other fields, such as the fields of $p$-adic numbers or for absolute or approximate factorization for instance. The core of our reduction uses the Newton polygon of the input polynomial, and its validity is based on the Newton-Puiseux expansion of roots of bivariate polynomials. In particular, we bound the valuation of $f(X,\phi)$ where $f$ is a lacunary polynomial and $\phi$ a Puiseux series whose vanishing polynomial has low degree.Comment: 22 page

A lifting and recombination algorithm for rational factorization of sparse polynomials

We propose a new lifting and recombination scheme for rational bivariate polynomial factorization that takes advantage of the Newton polytope geometry. We obtain a deterministic algorithm that can be seen as a sparse version of an algorithm of Lecerf, with now a polynomial complexity in the volume of the Newton polytope. We adopt a geometrical point of view, the main tool being derived from some algebraic osculation criterions in toric varieties.Comment: 22 page

Computation of Atomic Fibers of Z-Linear Maps

For given matrix $A\in\Z^{d\times n}$, the set $P_{b}=\{z:Az=b,z\in\Z^n_+\}$ describes the preimage or fiber of $b\in\Z^d$ under the $\Z$-linear map $f_A:\Z^n_+\to\Z^d$, $x\mapsto Ax$. The fiber $P_{b}$ is called atomic, if $P_{b}=P_{b_1}+P_{b_2}$ implies $b=b_1$ or $b=b_2$. In this paper we present a novel algorithm to compute such atomic fibers. An algorithmic solution to appearing subproblems, computational examples and applications are included as well.Comment: 27 page

Towards Toric Absolute Factorization

International audienceThis article gives an algorithm to recover the absolute factorization of a bivariate polynomial, taking into account the geometry of its monomials. It is based on algebraic criterions inherited from algebraic interpolation and toric geometry

Computation of atomic fibers of Z-linear maps

For given matrix $A\in\Z^{d\times n}$, the set $P_{b}=\{z:Az=b,z\in\Z^n_+\}$ describes the preimage or fiber of $b\in\Z^d$ under the $\Z$-linear map $f_A:\Z^n_+\rightarrow\Z^d$, $x\mapsto Ax$. The fiber $P_{b}$ is called atomic, if $P_{b}=P_{b_1}+P_{b_2}$ implies $b=b_1$ or $b=b_2$. In this paper we present a novel algorithm to compute such atomic fibers. An algorithmic solution to appearing subproblems, computational examples and applications are included as well

Factoring polynomials via polytopes

We introduce a new approach to multivariate polynomial factorisation which incorporates ideas from polyhedral geometry, and generalises Hensel lifting. Our main contribution is to present an algorithm for factoring bivariate polynomials which is able to exploit to some extent the sparsity of polynomials. We give details of an implementation which we used to factor randomly chosen sparse and composite polynomials of high degree over the binary field. Copyright 2004 ACM

Factoring polynomials via polytopes

We introduce a new approach to multivariate polynomial factorisation which incorporates ideas from polyhedral geometry, and generalises Hensel lifting. Our main contribution is to present an algorithm for factoring bivariate polynomials which is able to exploit to some extent the sparsity of polynomials. We give details of an implementation which we used to factor randomly chosen sparse and composite polynomials of high degree over the binary field

Factoring Polynomials via Polytopes

We introduce a new approach to multivariate polynomial factorisation which incorporates ideas from polyhedral geometry, and generalises Hensel lifting. Our main contribution is to present an algorithm for factoring bivariate polynomials which is able to exploit to some extent the sparsity of polynomials. We give details of an implementation which we used to factor randomly chosen sparse and composite polynomials of high degree over the binary field

An efficient sparse adaptation of the polytope method over Fp and a record-high binary bivariate factorisation

AbstractA recent bivariate factorisation algorithm appeared in Abu-Salem et al. [Abu-Salem, F., Gao, S., Lauder, A., 2004. Factoring polynomials via polytopes. In: Proc. ISSAC’04. pp. 4–11] based on the use of Newton polytopes and a generalisation of Hensel lifting. Although possessing a worst-case exponential running time like the Hensel lifting algorithm, the polytope method should perform well for sparse polynomials whose Newton polytopes have very few Minkowski decompositions. A preliminary implementation in Abu-Salem et al. [Abu-Salem, F., Gao, S., Lauder, A., 2004. Factoring polynomials via polytopes. In: Proc. ISSAC’04. pp. 4–11] indeed reflects this property, but does not exploit the fact that the algorithm preserves the sparsity of the input polynomial, so that the total amount of work and space required are O(d4) and O(d2) respectively, for an input bivariate polynomial of total degree d. In this paper, we show that the polytope method can be made sensitive to the number of non-zero terms of the input polynomial, so that the input size becomes dependent on both the degree and the number of terms of the input bivariate polynomial. We describe a sparse adaptation of the polytope method over finite fields with prime order, which requires fewer bit operations and memory references given a degree d sparse polynomial whose number of terms t satisfies t<d3/4, and which is known to be the product of two sparse factors. For t<d, and using fast polynomial arithmetic over finite fields, our refinement reduces the amount of work per extension of a coprime dominating edge factorisation and the total spatial cost to O(tλd2+t2λdL(d)+t4λd) bit operations and O(tλd) bits of memory respectively, for some 1/2≤λ<1, and L(d)=logdloglogd. To the best of our knowledge, the sparse binary factorisations achieved using this adaptation are of the highest degree so far, reaching a world record degree of 20000 for a very sparse bivariate polynomial over F2