# Geometry of nondegenerate polynomials: Motivic nearby cycles and Cohomology of contact loci

## TL;DR

This paper investigates the geometric and cohomological properties of nondegenerate polynomials using motivic nearby cycles and contact loci, providing explicit descriptions and partial answers to conjectures in the field.

## Contribution

It introduces explicit descriptions of motivic nearby cycles and contact loci for nondegenerate polynomials, advancing understanding of their geometric and cohomological structures.

## Key findings

- Explicit formulas for motivic nearby cycles of nondegenerate polynomials
- Partial resolution of the integral identity conjecture in this context
- Cohomology calculations for contact loci of Kouchnirenko nondegenerate polynomials

## Abstract

We study polynomials with complex coefficients which are nondegenerate in two senses, one of Kouchnirenko and the other with respect to its Newton polyhedron, through data on contact loci and motivic nearby cycles. Introducing an explicit description of these quantities we can answer in part to questions concerning the motivic nearby cycles of restriction functions and the integral identity conjecture in the context of Newton nondegenerate polynomials. Furthermore, in the nondegeneracy in the sense of Kouchnirenko, we give calculations on cohomology groups of the contact loci.