Exponential lower bound via exponential sums

Valiant’s famous VP vs. VNP conjecture states that the symbolic permanent polynomial does not have polynomial-size algebraic circuits. However, the best upper bound on the size of the circuits computing the permanent is exponential. Informally, VNP is an exponential sum of VP-circuits. In this paper we study whether, in general, exponential sums (of algebraic circuits) require exponential-size algebraic circuits. We show that the famous Shub-Smale $τ$-conjecture indeed implies such an exponential lower bound for an exponential sum. Our main tools come from parameterized complexity. Along the way, we also prove an exponential fpt (fixed-parameter tractable) lower bound for the parameterized algebraic complexity class VW${nb}^0$[P], assuming the same conjecture. VW${nb}^0$[P] can be thought of as the weighted sums of (unbounded-degree) circuits, where only $\pm 1$ constants are cost-free. To the best of our knowledge, this is the first time the Shub-Smale $τ$-conjecture has been applied to prove explicit exponential lower bounds. Furthermore, we prove that when this class is fpt, then a variant of the counting hierarchy, namely the linear counting hierarchy collapses. Moreover, if a certain type of parameterized exponential sums is fpt, then integers, as well as polynomials with coefficients being definable in the linear counting hierarchy have subpolynomial $τ$-complexity. Finally, we characterize a related class VW[F], in terms of permanents, where we consider an exponential sum of algebraic formulas instead of circuits. We show that when we sum over cycle covers that have one long cycle and all other cycles have constant length, then the resulting family of polynomials is complete for VW[F] on certain types of graphs.

💡 Research Summary

The paper investigates whether exponential sums of algebraic circuits—precisely the objects that define the class VNP—necessarily require exponential‑size algebraic circuits. While VNP is known to be captured by exponential sums of VP‑circuits, the best known upper bound for the permanent (the canonical VNP‑complete polynomial) remains exponential, and it is open whether sub‑exponential circuits exist. The authors connect this question to the Shub‑Smale τ‑conjecture, which posits that the number of integer zeros of a polynomial is polynomially bounded in its τ‑complexity (the size of the smallest division‑free circuit computing it from ±1).

The central technical construct is the parameterized problem p‑log‑Expsum(m,k,g). Given a circuit g of size m with n input variables X and ℓ=O(n) auxiliary Boolean variables Y, the problem asks for the exponential sum ∑_{y∈{0,1}^ℓ} g(X,y), where the parameter k = n / log m. The authors prove two complementary statements:

1. If p‑log‑Expsum were fixed‑parameter tractable (fpt) with respect to k, then the linear counting hierarchy (CH_lin) would collapse, and every integer or univariate polynomial whose bits are computable in CH_lin would have sub‑polynomial τ‑complexity. This is shown by encoding the counting hierarchy’s quantifier structure into exponential sums and using the assumed fpt algorithm to obtain sub‑exponential constant‑free circuits for characteristic functions of higher‑level CH languages.

2. Assuming the τ‑conjecture, p‑log‑Expsum cannot be fpt. The proof uses the Pochhammer polynomial p_n(x)=∏_{i=1}^n (x+i). Its coefficient of x^{n−k} equals the elementary symmetric polynomial σ_k(1,…,n). The sequence σ_k(1,…,n) is definable in CH_lin, so under the τ‑conjecture its τ‑complexity must be super‑polynomial. If p‑log‑Expsum were fpt, σ_k would admit constant‑free circuits of size 2^{o(n)}, contradicting the τ‑conjecture. Consequently, there exists an explicit VNP‑family (the exponential sum defined by p‑log‑Expsum) that requires circuits of size 2^{Ω(ℓ)}.

Beyond this core result, the paper introduces parameterized algebraic complexity classes VW_nb