How often do polynomials hit squares?

Abstract: Given a polynomial with rational coefficients, one may investigate the possible values that may be attained by these polynomials over the set of rational numbers. For centuries, number theorists have been giving due attention to square rational values assumed by rational polynomials. It turns out that seeking an answer to this question connects number theory and geometry. Answering this question for a polynomial in one variable will lead us to study the arithmetic of certain algebraic curves. We will spend some time explaining the geometry beneath the question when the degree of the polynomial is at least 3. For polynomials in more than one variable, the geometric structure is remote. In the latter case, we will present some of the old and recent developments in the theory shedding some light on some classical Diophantine questions.

algebraic geometrynumber theory

Audience: researchers in the topic

Comments: Zoom link: us06web.zoom.us/j/85613860958 Meeting ID: 856 1386 0958

FGC-HRI-IPM Number Theory Webinars

Series comments: password is 848084