如果 VNP 是困难的，那么它的方程也是困难的
If VNP is hard, then so are equations for it
AI总结：
本文在 Permanent 多项式需要指数规模代数电路的假设下，证明 VNP 类不存在高效可计算的方程，从而说明允许大系数多项式会使方程的电路复杂度显著膨胀。
摘要：
假设 Permanent 多项式需要指数规模的代数电路，我们证明类 VNP 不具有高效可计算的方程。换句话说，任何在所有 VNP 类多项式的系数向量上都为零的非零多项式，都需要超多项式规模的代数电路。在 Chatterjee 与作者最近的工作（FOCS 2020）中，已经证明由具有有界整数系数的多项式组成的 VP 和 VNP 子类确实具有小代数电路的方程。他们的工作留下了这些结果或许可以扩展到整个 VP 或 VNP 的可能性。本文的结果表明，在假设 Permanent 困难的前提下，至少对于 VNP 而言，允许具有大系数的多项式确实会导致方程的电路复杂度显著膨胀。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
