Measuring Singularities with Frobenius: The Basics

Consider a polynomial $f$ defined over a field $k$, the multiplicity is perhaps the most naive measurement of the singularities of $f$. This paper describes the first steps toward understanding a much more subtle measure of singularities which arises naturally in three different contexts-- analytic, algebro-geometric, and finally, algebraic. Miraculously, all three approaches lead to essentially the same measurement of singularities: the log canonical threshold (in characteristic zero) and the closely related $F$-pure threshold (in characteristic $p$). In this paper we present only the first steps in understanding these invariants, with an emphasis on the prime characteristic setting.