Schemic Grothendieck rings and motivic rationality

TL;DR

This paper introduces the schemic Grothendieck ring as a refined invariant that captures non-reduced structures and enables a motivic integration framework, leading to new proofs of rationality results in algebraic geometry.

Contribution

It develops the schemic Grothendieck ring, extending classical invariants, and provides a characteristic-free proof of the rationality of Igusa zeta series for certain hypersurfaces.

Findings

Defined the schemic Grothendieck ring incorporating non-reduced schemes.

Established a motivic integration approach using arc schemes.

Proved the rationality of geometric Igusa zeta series for specific hypersurfaces.

Abstract

We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme is represented, while maintaining its non-reduced structure. This yields a more subtle invariant, called the schemic Grothendieck ring, in which we can formulate a form of integration resembling Kontsevich's motivic integration via arc schemes. In view of its more functorial properties, we can present a characteristic-free proof of the rationality of the geometric Igusa zeta series for certain hypersurfaces, thus generalizing the ground-breaking work on motivic integration by Denef and Loeser. The construction uses first-order formulae, and some infinitary versions, called formularies.