通过电路判别式实现更快的实际可行性
Faster Real Feasibility via Circuit Discriminants
摘要：
我们证明了对于任意固定的n，在稀疏编码中，可以在时间多项式中检测诚实的n-变量（n+2）-范数（具有整数指数和系数）的实根。在稀疏编码中，即使n是固定的，以前最好的复杂度界限也是指数的。然后我们给出了这些函数k（n）的一个特征，使得当n趋于无穷大时，检测n元（n+k（n））-范数的实根的复杂性从P过渡到NP-困难。我们的证明遵循在很大程度上从一个新的复杂性阈值，决定消失的A-判别的n-变量（n+k（n））-nomials。丢番图逼近，通过线性型在矩阵中，也出现作为一个关键的工具。
英文摘要：
We show that detecting real roots for honestly n-variate (n+2)-nomials (with integer exponents and coefficients) can be done in time polynomial in the sparse encoding for any fixed n. The best previous complexity bounds were exponential in the sparse encoding, even for n fixed. We then give a characterization of those functions k(n) such that the complexity of detecting real roots for n-variate (n+k(n))-nomials transitions from P to NP-hardness as n tends to infinity. Our proofs follow in large part from a new complexity threshold for deciding the vanishing of A-discriminants of n-variate (n+k(n))-nomials. Diophantine approximation, through linear forms in logarithms, also arises as a key tool.
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