A generic effective Oppenheim theorem for systems of forms

TL;DR

This paper establishes a uniform effective density theorem and an effective counting result for generic systems of polynomial and linear forms, utilizing advanced techniques from the geometry of numbers and lattice theory.

Contribution

It introduces a new uniform effective density theorem for systems of forms, extending previous results to more general and generic settings.

Findings

Proves a uniform effective density theorem for systems of forms.

Provides an effective counting result for these systems.

Utilizes Roger's second moment formula on the space of unimodular lattices.

Abstract

We prove a uniform effective density theorem as well as an effective counting result for a generic system comprising a polynomial with a mild homogeneous condition and several linear forms using Roger's second moment formula for the Siegel transform on the space of unimodular lattices.