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 SIGMA, 2018, Volume 14, 074, 28 pp. (Mi sigma1373)

On Regularization of Second Kind Integrals

Julia Bernatskaa, Dmitry Leykinb

a National University of Kyiv-Mohyla Academy, 2 H. Skovorody Str., 04655 Kyiv, Ukraine
b 44/2 Harmatna Str., apt. 32, 03067 Kyiv, Ukraine

Abstract: We obtain expressions for second kind integrals on non-hyperelliptic $(n,s)$-curves. Such a curve possesses a Weierstrass point at infinity which is a branch point where all sheets of the curve come together. The infinity serves as the basepoint for Abel's map, and the basepoint in the definition of the second kind integrals. We define second kind differentials as having a pole at the infinity, therefore the second kind integrals need to be regularized. We propose the regularization consistent with the structure of the field of Abelian functions on Jacobian of the curve. In this connection we introduce the notion of regularization constant, a uniquely defined free term in the expansion of the second kind integral over a local parameter in the vicinity of the infinity. This is a vector with components depending on parameters of the curve, the number of components is equal to genus of the curve. Presence of the term guarantees consistency of all relations between Abelian functions constructed with the help of the second kind integrals. We propose two methods of calculating the regularization constant, and obtain these constants for $(3,4)$, $(3,5)$, $(3,7)$, and $(4,5)$-curves. By the example of $(3,4)$-curve, we extend the proposed regularization to the case of second kind integrals with the pole at an arbitrary fixed point. Finally, we propose a scheme of obtaining addition formulas, where the second kind integrals, including the proper regularization constants, are used.

Keywords: second kind integral; regularization constant; Abelian function relation; Jacobi inversion problem; addition formula.

DOI: https://doi.org/10.3842/SIGMA.2018.074

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MSC: 32A55; 32W50; 35R01; 14H40; 32A15; 14H45
Received: October 3, 2017; in final form July 2, 2018; Published online July 21, 2018
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Citation: Julia Bernatska, Dmitry Leykin, “On Regularization of Second Kind Integrals”, SIGMA, 14 (2018), 074, 28 pp.

Citation in format AMSBIB
\Bibitem{BerLei18} \by Julia~Bernatska, Dmitry~Leykin \paper On Regularization of Second Kind Integrals \jour SIGMA \yr 2018 \vol 14 \papernumber 074 \totalpages 28 \mathnet{http://mi.mathnet.ru/sigma1373} \crossref{https://doi.org/10.3842/SIGMA.2018.074} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000439656000001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85051851833}