拟阵、动机与Kontsevich猜想
Matroids, motives and conjecture of Kontsevich
摘要：
设G是有限连通图。G的Kirchhoff多项式是一个次数等于G的第一个贝蒂数的齐次多项式。这些多项式出现在电路的研究和费曼振幅的评估。受D. Kreimer和D. J. Broadhurst将多个zeta值与某些Feynman积分联系起来，Kontsevich证明了在具有q个元素的域上的基尔霍夫多项式的零点的数目总是q的多项式函数。通过将基尔霍夫多项式定义的方案与拟阵的表示空间联系起来，我们证明了这个猜想是错误的。利用Mnev的普适性定理，证明了这些方案本质上生成整数上所有有限型方案的算术.
英文摘要：
Let G be a finite connected graph. The Kirchhoff polynomial of G is a certain homogeneous polynomial whose degree is equal to the first betti number of G. These polynomials appear in the study of electrical circuits and in the evaluation of Feynman amplitudes. Motivated by work of D. Kreimer and D. J. Broadhurst associating multiple zeta values to certain Feynman integrals, Kontsevich conjectured that the number of zeros of a Kirchhoff polynomial over the field with q elements is always a polynomial function of q. We show that this conjecture is false by relating the schemes defined by Kirchhoff polynomials to the representation spaces of matroids. Moreover, using Mnev's universality theorem, we show that these schemes essentially generate all arithmetic of schemes of finite type over the integers.
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