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arXiv · 2007.00213

On the coefficient-choosing game

Divyum Sharma·L. Singhal
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Abstract

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 $\mathbb{Q}$.

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Divyum Sharma, L. Singhal. 2020-07-01. On the coefficient-choosing game. https://doi.org/10.2140/moscow.2021.10.183

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