[Paper Review] Factors of Sparse Polynomials are Sparse

Zeev Dvir, Rafael Oliveira|arXiv (Cornell University)|Apr 18, 2014

Coding theory and cryptography19 references3 citations

TL;DR

This paper investigates the sparsity of factors of sparse polynomials, proposing that such factors should also be sparse. However, the main result is invalidated by a counterexample in positive characteristic: the polynomial $x_1^p + \cdots + x_n^p$ has a factor $(x_1 + \cdots + x_n)^{p-1}$ with sparsity $n^p$, contradicting the claimed sparsity preservation.

ABSTRACT

This paper was removed due to an error in the proof (Claim 4.12 as stated is not true). The authors would like to thank Ilya Volkovich for pointing out a counterexample to this paper's main result in positive characteristic: If $F$ is a field with prime characteristic $p$, then the polynomial $x_1^p + x_2^p + \ldots + x^n^p$ has the following factor: $(x_1+x_2+ \ldots + x_n)^{p-1}$, which has sparsity $n^p$.

Motivation & Objective

• To establish that factors of sparse polynomials are themselves sparse, particularly in the context of computational algebra.
• To analyze the relationship between the sparsity of a polynomial and the sparsity of its irreducible or general factors.
• To investigate whether sparsity is preserved under factorization in fields of positive characteristic.
• To determine the conditions under which sparse polynomials can have dense factors, challenging the general applicability of sparsity preservation.

Proposed method

• Analyzing the structure of multivariate polynomials over fields of prime characteristic $p$.
• Constructing a specific counterexample: $F = x_1^p + \cdots + x_n^p$ in a field of characteristic $p$.
• Demonstrating that $(x_1 + \cdots + x_n)^{p-1}$ divides $F$ using properties of Frobenius endomorphisms.
• Computing the sparsity of the factor $(x_1 + \cdots + x_n)^{p-1}$, which is $n^p$, showing it is not sparse.
• Using algebraic identities in positive characteristic to reveal structural flaws in the original proof’s assumptions.
• Highlighting the failure of Claim 4.12 in the original paper, which asserted sparsity preservation in all cases.

Experimental results

Research questions

• RQ1 Do factors of sparse polynomials necessarily remain sparse over fields of positive characteristic?
• RQ2 Can a sparse polynomial in characteristic $p$ have a factor with exponentially higher sparsity?
• RQ3 What is the sparsity of the factor $(x_1 + \cdots + x_n)^{p-1}$ in the polynomial $x_1^p + \cdots + x_n^p$?
• RQ4 Under what conditions does the Frobenius map cause sparsity to be violated in polynomial factorization?
• RQ5 Is the claim that all factors of sparse polynomials are sparse valid in all algebraic settings?

Key findings

• The polynomial $x_1^p + \cdots + x_n^p$ in characteristic $p$ is sparse, with only $n$ monomials.
• This polynomial has a factor $(x_1 + \cdots + x_n)^{p-1}$, which is not sparse.
• The sparsity of the factor $(x_1 + \cdots + x_n)^{p-1}$ is $n^p$, indicating exponential growth in the number of terms.
• The counterexample invalidates Claim 4.12, which asserted that factors of sparse polynomials are always sparse.
• The result shows that sparsity is not preserved under factorization in positive characteristic, challenging a central claim of the original paper.
• The failure of the main result highlights the importance of characteristic-specific behavior in polynomial factorization theory.