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

An exponential sum is a linear combination of characters of the additive group of $\mathbb C^n$. 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 zeros of 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 and apply the De Concini–Procesi algorithm to construct the ring of conditions of the corresponding intersection theory. To construct the intersection number and the ring of conditions, we associate an algebraic subvariety of a multidimensional complex torus with every $\mathrm{EA}$-set and use the methods of tropical geometry. By computing the intersection number of the divisors of arbitrary exponential sums $f_1,\dots,f_n$, we arrive at a formula for the density of the $\mathrm{EA}$-set of common zeros of the perturbed system $f_i(z+w_i)$, where the perturbation $\{w_1,\dots,w_n\}$ belongs to a set of relatively full measure in $\mathbb{C}^{n\times n}$. This formula is analogous to the formula for the number of common zeros of Laurent polynomials.