Volume 5 (2022)
Roques, Julien; Singer, Michael F.
On the algebraic dependence of holonomic functions
Annales Henri Lebesgue, Volume 5 (2022), pp. 141-177.

Metadata

DOI10.5802/ahl.120
Keywords Linear Differential Equations, Differential Galois Theory, Algebraic Relations, Iterated Integrals, Hypergeometric Functions

Abstract

We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular, if f and g satisfy linear differential equations and are algebraically dependent, we give conditions on the differential Galois group associated to f guaranteeing that g is a polynomial in f. We apply this to hypergeometric functions and iterated integrals.


References

[BBH88] Beukers, Frits; Brownawell, W. Dale; Heckman, Gert Siegel normality, Ann. Math., Volume 127 (1988) no. 2, pp. 279-308 | DOI | MR | Zbl

[BH89] Beukers, Frits; Heckman, Gert Monodromy for the hypergeometric function n F n-1 , Invent. Math., Volume 95 (1989) no. 2, pp. 325-354 | DOI | MR | Zbl

[Bor91] Borel, Armand Linear algebraic groups, Graduate Texts in Mathematics, 126, Springer, 1991 | DOI | MR | Zbl

[DM81] Deligne, Pierre; Milne, James S. Tannakian Categories, Hodge Cycles, Motives, and Shimura Varieties (Lecture Notes in Mathematics), Volume 900, Springer, 1981, pp. 101-228 | DOI | Zbl

[DM89] Duval, Anne; Mitschi, Claude Matrices de Stokes et groupe de Galois des équations hypergéométriques confluentes généralisées, Pac. J. Math., Volume 138 (1989) no. 1, pp. 25-56 | DOI | MR | Zbl

[Hoc81] Hochschild, Gerhard P. Basic theory of algebraic groups and Lie algebras, Graduate Texts in Mathematics, 75, Springer, 1981 | DOI | MR | Zbl

[HS85] Harris Jr., William A.; Sibuya, Yasataka The reciprocals of solutions of linear ordinary differential equations, Adv. Math., Volume 58 (1985) no. 2, pp. 119-132 | DOI | MR | Zbl

[HS86] Harris Jr., William A.; Sibuya, Yasataka The n th roots of solutions of linear ordinary differential equations, Proc. Am. Math. Soc., Volume 97 (1986) no. 2, pp. 207-211 | DOI | MR | Zbl

[Hum75] Humphreys, James E. Linear algebraic groups, Graduate Texts in Mathematics, 21, Springer, 1975 | MR | Zbl

[Kat72] Katz, Nicholas M. Algebraic solutions of differential equations (p-curvature and the Hodge filtration), Invent. Math., Volume 18 (1972), pp. 1-118 | DOI | MR | Zbl

[Kat87] Katz, Nicholas M. On the calculation of some differential Galois groups, Invent. Math., Volume 87 (1987) no. 1, pp. 13-61 | DOI | MR | Zbl

[Kat90] Katz, Nicholas M. Exponential sums and differential equations, Annals of Mathematics Studies, 124, Princeton University Press, 1990 | DOI | MR | Zbl

[Kol68] Kolchin, Ellis R. Algebraic groups and algebraic dependence, Am. J. Math., Volume 90 (1968), pp. 1151-1164 | DOI | MR | Zbl

[Mag94] Magid, Andy R. Lectures on differential Galois theory, University Lecture Series, 7, American Mathematical Society, 1994 | Zbl

[Mit96] Mitschi, Claude Differential Galois groups of confluent generalized hypergeometric equations: an approach using Stokes multipliers, Pac. J. Math., Volume 176 (1996) no. 2, pp. 365-405 | DOI | MR | Zbl

[NvdPT08] Nguyen, An Khuong; van der Put, Marius; Top, Jaap Algebraic subgroups of GL 2 (ℂ), Indag. Math., New Ser., Volume 19 (2008) no. 2, pp. 287-297 | DOI | MR | Zbl

[PS03] van der Put, Marius; Singer, Michael F. Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer, 2003 | DOI | MR | Zbl

[Roq14] Roques, Julien On generalized hypergeometric equations and mirror maps, Proc. Am. Math. Soc., Volume 142 (2014) no. 9, pp. 3153-3167 | DOI | MR | Zbl

[Sin86] Singer, Michael F. Algebraic relations among solutions of linear differential equations, Trans. Am. Math. Soc., Volume 295 (1986) no. 2, pp. 753-763 | DOI | MR | Zbl

[Spe86] Sperber, Steven On solutions of differential equations which satisfy certain algebraic relations, Pac. J. Math., Volume 124 (1986) no. 1, pp. 249-256 | DOI | MR | Zbl

[Sri10] Srinivasan, Varadharaj R. Iterated antiderivative extensions, J. Algebra, Volume 324 (2010) no. 8, pp. 2042-2051 | DOI | MR | Zbl