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Izvestiya: Mathematics, 2022, Volume 86, Issue 1, Pages 169–202
DOI: https://doi.org/10.1070/IM9065
(Mi im9065)
 

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
References:
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.
Received: 25.05.2020
Revised: 09.10.2020
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2022, Volume 86, Issue 1, Pages 180–218
DOI: https://doi.org/10.4213/im9065
Bibliographic databases:
Document Type: Article
UDC: 512.734+512.816.7+517.550.4
MSC: 14T05, 14C17, 52A30
Language: English
Original paper language: Russian
Citation: B. Ya. Kazarnovskii, “The quasi-algebraic ring of conditions of $\mathbb C^n$”, Izv. RAN. Ser. Mat., 86:1 (2022), 180–218; Izv. Math., 86:1 (2022), 169–202
Citation in format AMSBIB
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\paper The quasi-algebraic ring of~conditions of~$\mathbb C^n$
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\vol 86
\issue 1
\pages 180--218
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    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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