论文标题
在系数选择游戏中
On the coefficient-choosing game
论文作者
论文摘要
诺拉（Nora）和万达（Wanda）是两个球员，他们从某些固定的Unitual“交换环” $ r $中选择$ d $多项式的系数。如果多项式在$ r $的分数中有根源，那么万达将被宣布为获胜者，而诺拉则被宣布为获胜者。我们将Gasarch，Washington和Zbarsky给出的这些游戏的理论扩展到所有有限的循环环，并确定可能的结果。还使用离散评估环为这些作者提出的游戏的变体构建了一个示例家族。我们那里的技术使我们采用了一种对抗性方法，用于构建任何规定学位的理性多项式（等于$ 3 $或大于$ 8 $），而在最大的Abelian扩展中没有$ \ mathbb {q} $的最大阳性。
Nora and Wanda are two players who choose coefficients of a degree $d$ polynomial from some fixed unital commutative ring $R$. Wanda is declared the winner if the polynomial has a root in the ring of fractions of $R$ and Nora is declared the winner otherwise. We extend the theory of these games given by Gasarch, Washington and Zbarsky to all finite cyclic rings and determine the possible outcomes. A family of examples is also constructed using discrete valuation rings for a variant of the game proposed by these authors. Our techniques there lead us to an adversarial approach to constructing rational polynomials of any prescribed degree (equal to $3$ or greater than $8$) with no roots in the maximal abelian extension of $\mathbb{Q}$.
