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2021 On the coefficient-choosing game
Divyum Sharma, L. Singhal
Mosc. J. Comb. Number Theory 10(3): 183-202 (2021). DOI: 10.2140/moscow.2021.10.183

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 (2018) 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 .

Citation

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Divyum Sharma. L. Singhal. "On the coefficient-choosing game." Mosc. J. Comb. Number Theory 10 (3) 183 - 202, 2021. https://doi.org/10.2140/moscow.2021.10.183

Information

Received: 29 November 2020; Revised: 15 June 2021; Accepted: 5 July 2021; Published: 2021
First available in Project Euclid: 15 November 2021

Digital Object Identifier: 10.2140/moscow.2021.10.183

Subjects:
Primary: 91A46
Secondary: 11C08 , 11S05

Keywords: finite cyclic rings , Newton polygons , roots of polynomials

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.10 • No. 3 • 2021
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