Algebraic and modular properties of hypergeometric functions

src/doc/en/reference/functions/index.rst

@@ -21,6 +21,7 @@ Built-in Functions
    sage/functions/other
    sage/functions/special
    sage/functions/hypergeometric
+   sage/functions/hypergeometric_algebraic
    sage/functions/jacobi
    sage/functions/airy
    sage/functions/bessel

src/doc/en/reference/references/index.rst

@@ -1432,6 +1432,9 @@ REFERENCES:
               An Algorithmic Approach, Algorithms and Computation in Mathematics,
               Volume 20, Springer (2007)
 
+.. [But2010] Peter Butkovič, *Max-linear systems. Theory and algorithms.*
+             Springer Monographs in Mathematics. London: Springer. xvii, 272 p. (2010). 
+
 .. [Buell89] Duncan A. Buell.
              *Binary Quadratic Forms: Classical Theory and Modern Computations.*
              Springer, 1989.
@@ -1650,6 +1653,10 @@ REFERENCES:
              IV. The quotient groups of the lower central series, Ann. of Math.
              68 (1958) 81--95.
 
+.. [CFV2025] Xavier Caruso, Florian Fürnsinn, Daniel Vargas-Montoya,
+           *Galois groups of reductions modulo p of D-finite series*,
+           :arxiv:`2504.09429`, 2025.
+
 .. [CFZ2000] \J. Cassaigne, S. Ferenczi, L.Q. Zamboni, *Imbalances in
              Arnoux-Rauzy sequences*, Ann. Inst. Fourier (Grenoble)
              50 (2000) 1265--1276.
@@ -1724,6 +1731,9 @@ REFERENCES:
 .. [ChenDB] Eric Chen, Online database of two-weight codes,
             http://moodle.tec.hkr.se/~chen/research/2-weight-codes/search.php
 
+.. [Chr1986] Gilles Christol, *Fonctions hypergéométriques bornées*.
+             Groupe de travail d’analyse ultramétrique 14 (1986-87), exp. no 8, p. 1-16
+
 .. [CHK2001] Keith D. Cooper, Timothy J. Harvey and Ken Kennedy. *A
              Simple, Fast Dominance Algorithm*, Software practice and
              Experience, 4:1-10 (2001).
@@ -2881,6 +2891,11 @@ REFERENCES:
             toric varieties defined by atomic lattices*. Inventiones
             Mathematicae. **155** (2004), no. 3, pp. 515-536.
 
+.. [FY2024] Florian Fürnsinn, Sergey Yurkevich. 
+            *Algebraicity of hypergeometric functions with arbitrary parameters*,
+            Bulletin of the London Mathematical Society. **56** (2024),
+            pp. 2824-2846. :doi:`10.1112/blms.13103`, :arxiv:`2308.12855` (2023).
+
 .. [FZ2001] \S. Fomin and A. Zelevinsky. *Cluster algebras I. Foundations*,
             \J. Amer. Math. Soc. **15** (2002), no. 2, pp. 497-529.
             :arxiv:`math/0104151` (2001).

src/doc/en/reference/semirings/index.rst

@@ -6,5 +6,9 @@ Standard Semirings
 
    sage/rings/semirings/non_negative_integer_semiring
    sage/rings/semirings/tropical_semiring
+   sage/rings/semirings/tropical_polynomial
+   sage/rings/semirings/tropical_mpolynomial
+   sage/rings/semirings/tropical_matrix
+   sage/rings/semirings/tropical_variety
 
 .. include:: ../footer.txt

src/sage/functions/hypergeometric.py

@@ -4,6 +4,10 @@
 This module implements manipulation of infinite hypergeometric series
 represented in standard parametric form (as `\,_pF_q` functions).
 
+For a more algebraic treatment of hypergeometric functions
+(including reduction modulo primes and `p`-adic properties),
+we refer to :mod:`sage.functions.hypergeometric_algebraic`.
+
 AUTHORS:
 
 - Fredrik Johansson (2010): initial version
@@ -301,10 +305,19 @@ def __call__(self, a, b, z, **kwargs):
             sage: hypergeometric([2, 3, 4], [4, 1], 1)
             hypergeometric((2, 3, 4), (4, 1), 1)
         """
-        return BuiltinFunction.__call__(self,
-                                        SR._force_pyobject(a),
-                                        SR._force_pyobject(b),
-                                        z, **kwargs)
+        from sage.rings.polynomial.polynomial_element import Polynomial
+        from sage.rings.power_series_ring_element import PowerSeries
+        if isinstance(z, (Polynomial, PowerSeries)):
+            if not z.is_gen():
+                raise NotImplementedError("the argument must be the generator of the polynomial ring")
+            S = z.parent()
+            from sage.functions.hypergeometric_algebraic import HypergeometricFunctions
+            return HypergeometricFunctions(S.base_ring(), S.variable_name())(a, b)
+        else:
+            return BuiltinFunction.__call__(self,
+                                            SR._force_pyobject(a),
+                                            SR._force_pyobject(b),
+                                            z, **kwargs)
 
     def _print_latex_(self, a, b, z):
         r"""