Vol. 10, No. 3, 2021

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On the coefficient-choosing game

Divyum Sharma and L. Singhal

Vol. 10 (2021), No. 3, 183–202
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 .

Keywords
roots of polynomials, finite cyclic rings, Newton polygons
Mathematical Subject Classification
Primary: 91A46
Secondary: 11C08, 11S05
Milestones
Received: 29 November 2020
Revised: 15 June 2021
Accepted: 5 July 2021
Published: 13 September 2021
Authors
Divyum Sharma
Department of Mathematics
Birla Institute of Technology and Science
Pilani
India
L. Singhal
Yau Mathematical Sciences Center
Tsinghua University
Beijing
China