[Paper Review] Closure of VP under taking factors: a short and simple proof

Chi-Ning Chou, Mrinal Kumar+1 more|arXiv (Cornell University)|Mar 6, 2019

semigroups and automata theory12 references4 citations

TL;DR

This paper presents a concise, self-contained proof that the complexity class VP is closed under taking factors, showing that any factor of a polynomial in VP can be computed by a polynomial-size arithmetic circuit. The proof leverages a multivariate Newton iteration and the resultant to directly reconstruct factors, avoiding the need for advanced tools like Hensel lifting or Hilbert's Irreducibility Theorem.

ABSTRACT

In this note, we give a short, simple and almost completely self contained proof of a classical result of Kaltofen [Kal86, Kal87, Kal89] which shows that if an $n$ variate degree $d$ polynomial $f$ can be computed by an arithmetic circuit of size $s$, then each of its factors can be computed by an arithmetic circuit of size at most $ extsf{poly}\left(s, n, d ight)$. However, unlike Kaltofen's argument, our proof does not directly give an efficient algorithm for computing the circuits for the factors of $f$.

Motivation & Objective

• To provide a simple and accessible proof of the classical result that VP is closed under taking factors.
• To replace Kaltofen's complex algebraic machinery—such as Hensel lifting and effective Hilbert's Irreducibility—with elementary techniques.
• To demonstrate that factor computation in VP can be achieved via iterative multivariate Newton approximation and GCD detection using the resultant.
• To offer a proof that is almost entirely self-contained and accessible to researchers without deep algebraic background.
• To highlight structural insights into factorization in algebraic complexity by simplifying the proof framework.

Proposed method

• Use a multivariate Newton iteration to iteratively approximate the homogeneous components of a factor of a polynomial.
• At each iteration, solve a system of linear equations derived from the condition that the product of approximated factors matches the original polynomial modulo higher-degree monomials.
• Employ the resultant of two univariate polynomials to detect whether they share a nontrivial GCD, enabling factor recovery.
• Construct circuits for the factor polynomials by lifting approximations degree-by-degree up to degree d.
• Homogenize the resulting circuit to recover the full homogeneous components of the factors.
• Use the decomposition f = z^e - g^e to reduce the case of repeated factors to the coprime case, then apply the main lemma.

Experimental results

Research questions

• RQ1 Can the closure of VP under factorization be proven without relying on deep algebraic tools like Hensel lifting or Hilbert's Irreducibility Theorem?
• RQ2 Is there a direct, iterative method to reconstruct factors of a polynomial using only Newton iteration and GCD detection via the resultant?
• RQ3 Can the proof of factor closure in VP be made significantly simpler and more accessible while retaining correctness?
• RQ4 Does a multivariate Newton iteration approach suffice to recover all factors of a polynomial given a small circuit for the original?
• RQ5 Can the repeated factor case be reduced to the coprime factor case using elementary algebraic transformations?

Key findings

• The paper establishes that any factor of a polynomial in VP can be computed by an arithmetic circuit of size at most poly(s, n, d), where s is the size of the circuit for the original polynomial.
• The proof avoids algorithmic construction of factors, focusing instead on existential closure: it shows that such circuits exist without providing an efficient construction algorithm.
• The multivariate Newton iteration technique allows for step-by-step reconstruction of factor components up to degree d, ensuring correctness modulo higher powers of the ideal ⟨x⟩.
• The use of the resultant enables detection of common factors between univariate polynomials, which is crucial for verifying GCD conditions during the iterative process.
• For the case of repeated factors f = g^e, the proof reduces the problem to the coprime case via the transformation f = z^e - g^e, then applies the main result to z - g.
• The final circuit for the factor g is constructed via homogenization and linear combinations of outputs, incurring only a poly(s, n, d) size blow-up.