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The quasi-algebraic ring of conditions of $\mathbb C^n$
B. Ya. Kazarnovskii Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
Abstract:
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.
Keywords:
exponential sum, intersection number, Newton polytope, tropical geometry.
UDC:
512.734+
512.816.7+
517.550.4
MSC: 14T05,
14C17,
52A30 Received: 25.05.2020
Revised: 09.10.2020
DOI:
10.4213/im9065
Fulltext:
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