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.

翻译：著名的Valiant VP与VNP猜想断言符号永久多项式不具有多项式规模的代数电路。然而，计算永久多项式所需电路规模的最佳上界是指数级的。非正式地说，VNP是VP电路的指数和。本文研究在一般情况下，代数电路的指数和是否必然需要指数规模的代数电路。我们证明著名的Shub-Smale $τ$-猜想确实蕴含了指数和的指数下界。我们的主要工具来自参数化复杂度理论。在此过程中，我们还基于同一猜想证明了参数化代数复杂度类VW$_{nb}^0$[P]的指数fpt（固定参数可处理）下界。VW$_{nb}^0$[P]可视为（无界次数的）电路的加权和，其中仅$\pm 1$常数为零成本。据我们所知，这是首次应用Shub-Smale $τ$-猜想证明显式的指数下界。此外，我们证明当该类为fpt时，计数层次结构的变体——线性计数层次结构会发生坍缩。进一步地，若某类参数化指数和为fpt，则整数以及系数在线性计数层次结构中可定义的多项式将具有次多项式$τ$-复杂度。最后，我们通过永久多项式刻画了相关类VW[F]，其中我们考虑代数公式而非电路的指数和。我们证明，当对具有一个长循环且其余循环长度恒定的循环覆盖求和时，所得多项式族在特定图类上对VW[F]是完全的。