An exponential sum is a linear combination of characters of the additive group of . We regard as an analogue of the torus , exponential sums as analogues of Laurent polynomials, and exponential analytic sets (-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 -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 -set and use the methods of tropical geometry. By computing the intersection number of the divisors of arbitrary exponential sums , we arrive at a formula for the density of the -set of common zeros of the perturbed system , where the perturbation belongs to a set of relatively full measure in . This formula is analogous to the formula for the number of common zeros of Laurent polynomials.

中文翻译：

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条件的拟代数环

指数和是 的加法群的特征的线性组合。我们将指数和视为环面的类似物，将指数和视为洛朗多项式的类似物，并将指数解析集（-集）（即指数和的有限系统的公共零点集）视为环面的代数子品种的类似物。利用这些类比，我们定义了 集合的交集数，并应用 De Concini–Procesi 算法来构造相应交集理论的条件环。为了构造交数和条件环，我们将多维复环面的代数子簇与每个集合相关联，并使用热带几何的方法。通过计算任意指数和的除数的交集数，我们得到了扰动系统的公共零点集合的密度公式，其中扰动属于 中的一组相对完整的测度。该公式类似于洛朗多项式的公共零点数量的公式。