The Coefficient-Choosing Game

TL;DR

This paper investigates a strategic game involving two players choosing polynomial coefficients over an integral domain, analyzing which player has a winning strategy based on the properties of the domain.

Contribution

It characterizes the winning strategies for the game over various integral domains, extending understanding of polynomial roots and game theory in algebraic structures.

Findings

Identifies domains where Wanda has a winning strategy.

Determines conditions under which Nora can guarantee a win.

Provides a classification of domains based on the game's outcome.

Abstract

Let $D$ be an integral domain. Two players, Nora and Wanda, alternately choose coefficients from $D$ for a polynomial of degree $d$. When they are done, if the polynomial has a root in the field of fractions of $D$, then Wanda wins. If not, then Nora wins. We determine, for many $D$, who wins this game.