If VNP is hard, then so are equations for it

TL;DR

This paper demonstrates that if the Permanent polynomial is hard to compute, then the class VNP lacks efficiently computable equations, indicating a fundamental complexity barrier for algebraic equations in VNP.

Contribution

It establishes that assuming the hardness of Permanent, polynomials with large coefficients in VNP require super-polynomial size circuits for their equations, extending previous results to the entire class.

Findings

VNP does not have efficiently computable equations under Permanent hardness assumption.

Polynomials with large coefficients in VNP require super-polynomial size algebraic circuits for equations.

Extends prior results on bounded coefficient subclasses to all of VNP.

Abstract

Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size. In a recent work of Chatterjee and the authors (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.