带约束实除子的实多项式 I. 基本群
Real polynomials with constrained real divisors. I. Fundamental groups
AI总结：
本研究拓展Arnold与Vassiliev的多项式空间拓扑研究，考虑实除子避开特定重数序列的实首一多项式空间，给出其基本群的显式表示、秩的上界与稳定性，并给出配边解释。
摘要：
20世纪80年代末，V.~Arnold与V.~Vassiliev开启了对次数为d、且实根重数均不超过给定正整数的实一元多项式空间的拓扑研究。在拓展他们的研究时，我们考虑了次数为d的实首一多项式空间，这类多项式的实除子避开取自给定合成偏序集的根重数序列，该偏序集在若干自然组合运算下封闭。本文聚焦这类空间的基本群。我们通过生成元与关系给出了基本群的显式表示，并证明在诸多情形下它们是自由群，其秩的上界为d的二次函数。我们还证明了当d足够大时，基本群具有稳定性。此外，我们证明这些基本群可解释为1-流形浸入圆柱S^1 × R的特殊配边，其像避开该偏序集中与圆柱生成元相关的相切模式。
英文摘要：
In the late 80s, V.~Arnold and V.~Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree d and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces of real monic univariate polynomials of degree d whose real divisors avoid sequences of root multiplicities taken from a given poset of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of such spaces. We find explicit presentations for the fundamental groups in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in d. We also show that the fundamental group stabilizes for d large. We further show that the fundamental groups admit an interpretation as special bordisms of immersions of 1-manifolds into the cylinder S^1 \times R, whose images avoid the tangency patterns from the poset with respect to the generators of the cylinder.
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