Georgia Institute of Technology College of Sciences

This talk will serve as an introduction to the algebra of hyperfields—fields with a multivalued addition. For example the sign hyperfield which is the arithmetic of real numbers modulo their absolute value (e.g. positive + positive = positive, positive + negative = any possibility). We will also introduce valued fields which capture the idea of how many times a fixed prime p divides the numerator or denominator of a rational number.

Using this arithmetic we will consider the combinatorial question of factoring a polynomial over a hyperfield. This will present a unified and conceptual way of looking at Descartes's rule of signs (how many positive roots does a real polynomial have) and the Newton polygon rule (how many roots are there which are divisible by p or p^2).