The multiplicative structure on polynomial continuous valuations

TL;DR

This paper introduces a new algebraic structure on polynomial smooth valuations, revealing a Frobenius algebra structure on translation invariant valuations and exploring its properties and applications.

Contribution

It establishes a canonical commutative associative filtered algebra structure on polynomial smooth valuations, with special properties on translation invariant valuations.

Findings

Polynomial smooth valuations form a commutative associative filtered algebra.

Translation invariant valuations have a Frobenius algebra structure.

Applications of the algebraic structure are discussed.

Abstract

We introduce a canonical structure of a commutative associative filtered algebra with the unit on polynomial smooth valuations, and study its properties. The induced structure on the subalgebra of translation invariant smooth valuations has especially nice properties (it is the structure of the Frobenius algebra). We also present some applications.