Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example, square roots) although the final result may be expressed without algebraic extensions. In this article, I present an improvement of this algorithm, which avoids algebraic extensions.
REFERENCES
1.
K.
Aomoto
, J. Math. Soc. Jpn.
27
, 248
(1975
).2.
K.
Matsumoto
, Kyushu J. Math.
48
, 335
(1994
).3.
K.
Cho
and K.
Matsumoto
, Nagoya Math. J.
139
, 67
(1995
).4.
K.
Matsumoto
, Osaka J. Math.
29
, 873
(1998
).5.
K.
Ohara
, Y.
Sugiki
, and N.
Takayama
, Funkcialaj Ekvacioj
46
, 213
(2003
).6.
7.
8.
9.
10.
11.
K.
Aomoto
and M.
Kita
, Theory of Hypergeometric Functions
(Springer
, 2011
).12.
13.
14.
15.
16.
17.
18.
S.
Mizera
, “Aspects of scattering amplitudes and moduli space localization
,” Ph.D. thesis, Perimeter Institute for Theoretical Physics
, 2019
; arXiv:1906.02099.19.
20.
21.
22.
23.
24.
25.
26.
27.
F. V.
Tkachov
, Phys. Lett. B
100
, 65
(1981
).28.
K. G.
Chetyrkin
and F. V.
Tkachov
, Nucl. Phys. B
192
, 159
(1981
).29.
K.
Mimachi
, K.
Ohara
, and M.
Yoshida
, Tohoku Math. J.
56
(4
), 531
(2004
).30.
31.
32.
33.
E.
Cattani
and A.
Dickenstein
, in Solving Polynomial Equations
, Algorithms and Computation in Mathematics Vol. 14, edited by M.
Bronstein
, et al. (Springer
, 2005
), p. 1
.34.
Y.
Zhang
, “Lecture notes on multi-loop integral reduction and applied algebraic geometry
,” arXiv:1612.02249 (2016
).35.
J.
Moser
, Math. Z.
72
, 379
(1959
).36.
37.
P.
Griffiths
and J.
Harris
, Principles of Algebraic Geometry
(John Wiley & Sons
, New York
, 1994
).38.
39.
40.
41.
42.
43.
44.
45.
46.
M.
Caffo
, H.
Czyz
, S.
Laporta
, and E.
Remiddi
, Nuovo Cimento A
111
, 365
(1998
); arXiv:hep-th/9805118.47.
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