The quasi-algebraic ring of conditions of $\mathbb C^n$

An exponential sum is a linear combination of characters of the additive groupof $\mathbb C^n$. We regard $\mathbb{C}^n$ as an analogue of the torus$(\mathbb{C}\setminus0)^n$, exponential sums as analogues of Laurent polynomials,and exponential analytic sets ($\mathrm{EA}$-sets), that is, the sets of common zerosof finite systems of exponential sums, as analogues of algebraic subvarieties of the torus.Using these analogies, we define the intersection number of $\mathrm{EA}$-sets andapply the De Concini–Procesi algorithm to construct the ring of conditions of the correspondingintersection theory. To construct the intersection number and the ring of conditions, weassociate an algebraic subvariety of a multidimensional complex torus with every$\mathrm{EA}$-set and use the methods of tropical geometry. By computing the intersectionnumber of the divisors of arbitrary exponential sums $f_1,…,f_n$, we arrive at a formulafor the density of the $\mathrm{EA}$-set of common zeros of the perturbed system $f_i(z+w_i)$,where the perturbation $\{w_1,…,w_n\}$ belongs to a set of relatively full measurein $\mathbb{C}^{n\times n}$. This formula is analogous to the formula for the numberof common zeros of Laurent polynomials.

exponential sum, intersection number, Newton polytope, tropical geometry