Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree

08/20/2018
by   Vishwas Bhargava, et al.
0

In this paper we study the problem of deterministic factorization of sparse polynomials. We show that if f ∈F[x_1,x_2,... ,x_n] is a polynomial with s monomials, with individual degrees of its variables bounded by d, then f can be deterministically factored in time s^poly(d) n. Prior to our work, the only efficient factoring algorithms known for this class of polynomials were randomized, and other than for the cases of d=1 and d=2, only exponential time deterministic factoring algorithms were known. A crucial ingredient in our proof is a quasi-polynomial sparsity bound for factors of sparse polynomials of bounded individual degree. In particular we show if f is an s-sparse polynomial in n variables, with individual degrees of its variables bounded by d, then the sparsity of each factor of f is bounded by s^O(d^2n). This is the first nontrivial bound on factor sparsity for d>2. Our sparsity bound uses techniques from convex geometry, such as the theory of Newton polytopes and an approximate version of the classical Carathéodory's Theorem. Our work addresses and partially answers a question of von zur Gathen and Kaltofen (JCSS 1985) who asked whether a quasi-polynomial bound holds for the sparsity of factors of sparse polynomials.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
03/15/2018

Some Closure Results for Polynomial Factorization and Applications

In a sequence of seminal results in the 80's, Kaltofen showed that the c...
research
08/07/2023

New Bounds on Quotient Polynomials with Applications to Exact Divisibility and Divisibility Testing of Sparse Polynomials

A sparse polynomial (also called a lacunary polynomial) is a polynomial ...
research
11/06/2019

How many zeros of a random sparse polynomial are real?

We investigate the number of real zeros of a univariate k-sparse polynom...
research
02/26/2015

Factorization of Motion Polynomials

In this paper, we consider the existence of a factorization of a monic, ...
research
07/22/2018

What Can (and Can't) we Do with Sparse Polynomials?

Simply put, a sparse polynomial is one whose zero coefficients are not e...
research
09/11/2017

Root Separation for Trinomials

We give a separation bound for the complex roots of a trinomial f ∈Z[X]....
research
06/01/2023

Attribute-Efficient PAC Learning of Low-Degree Polynomial Threshold Functions with Nasty Noise

The concept class of low-degree polynomial threshold functions (PTFs) pl...

Please sign up or login with your details

Forgot password? Click here to reset