等变格生成元与马尔可夫基
Equivariant lattice generators and Markov bases
摘要：
最近已经证明，单项式映射在一个大类关于无限对称群的作用，有对称性，双生成核。我们研究这类中最简单的非平凡族：由单个单项式给出的映射。考虑到相应的格映射，我们显式地构造了一个等变格生成集，其宽度（写下它所需的变量数）线性依赖于映射的宽度。这个结果是尖锐的，并显着改善了以前已知的上界，因为它不依赖于图像单项式的程度。在宽度为2的情况下，我们构造了一个明确的有限集的二项式生成的环面理想的对称。这个生成集的宽度和次数都严格地受单项式指数中的线性函数的限制。
英文摘要：
It has been shown recently that monomial maps in a large class respecting the action of the infinite symmetric group have, up to symmetry, finitely generated kernels. We study the simplest nontrivial family in this class: the maps given by a single monomial. Considering the corresponding lattice map, we explicitly construct an equivariant lattice generating set, whose width (the number of variables necessary to write it down) depends linearly on the width of the map. This result is sharp and improves dramatically the previously known upper bound as it does not depend on the degree of the image monomial. In the case of of width two, we construct an explicit finite set of binomials generating the toric ideal up to symmetry. Both width and degree of this generating set are sharply bounded by linear functions in the exponents of the monomial.
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