Ring of conditions for affine space

TL;DR

This paper develops a ring of conditions for affine space using exponential sums and analytic sets, extending concepts from algebraic geometry and tropical geometry to affine spaces.

Contribution

It constructs an analog of the ring of conditions for affine space based on exponential sums and tropical algebraic geometry, generalizing known results from algebraic tori.

Findings

The ring of conditions for affine space is generated by hypersurfaces.

Construction is based on exponential sums and tropical algebraic geometry.

Analogous to the ring of conditions for algebraic tori.

Abstract

The exponential sum (ES) is a linear combination of characters of an additive group $\mathbb C^n$. The exponential analytic set (EAS) is a set of common zeroes of a finite tuple of ESs. We consider ES and EAS as an analogs of Laurent polynomial and of algebraic variety in complex torus $(\mathbb{C}\setminus0)^n$. Respectively we construct the ring of conditions for $\mathbb C^n$ as an analog of the ring of conditions for $(\mathbb{C}\setminus0)^n$. The construction of this ring is based on the definition of associated to EAS algebraic subvariety of some multidimensional torus and on the applying tropical algebraic geometry to this subvariety. Just as in the case of a torus, the ring of conditions is generated by hypersurfaces. This preprint is an extended summary of the article proposed to "Izvestiya: Mathematics".