diff --git a/src/doc/en/reference/polynomial_rings/polynomial_rings_multivar.rst b/src/doc/en/reference/polynomial_rings/polynomial_rings_multivar.rst
index 46388f58f25..6147929ccf2 100644
--- a/src/doc/en/reference/polynomial_rings/polynomial_rings_multivar.rst
+++ b/src/doc/en/reference/polynomial_rings/polynomial_rings_multivar.rst
@@ -36,6 +36,8 @@ are implemented using the PolyBoRi library (cf. :mod:`sage.rings.polynomial.pbor
    sage/rings/polynomial/multi_polynomial_libsingular
    sage/rings/polynomial/multi_polynomial_ideal_libsingular
 
+   sage/rings/polynomial/msolve
+
    sage/rings/polynomial/polydict
    sage/rings/polynomial/hilbert
 
diff --git a/src/sage/features/msolve.py b/src/sage/features/msolve.py
index fa6679c2014..a7c7d5441b7 100644
--- a/src/sage/features/msolve.py
+++ b/src/sage/features/msolve.py
@@ -2,9 +2,11 @@
 r"""
 Feature for testing the presence of msolve
 
-`msolve <https://msolve.lip6.fr/>`_ is a multivariate polynomial system solver
-developed mainly by Jérémy Berthomieu (Sorbonne University), Christian Eder
-(TU Kaiserslautern), and Mohab Safey El Din (Sorbonne University).
+`msolve <https://msolve.lip6.fr/>`_ is a multivariate polynomial system solver.
+
+.. SEEALSO::
+
+    - :mod:`sage.rings.polynomial.msolve`
 """
 
 import subprocess
diff --git a/src/sage/rings/polynomial/msolve.py b/src/sage/rings/polynomial/msolve.py
index 2dafacbba66..65ae859f751 100644
--- a/src/sage/rings/polynomial/msolve.py
+++ b/src/sage/rings/polynomial/msolve.py
@@ -3,19 +3,17 @@
 Solution of polynomial systems using msolve
 
 `msolve <https://msolve.lip6.fr/>`_ is a multivariate polynomial system solver
-developed mainly by Jérémy Berthomieu (Sorbonne University), Christian Eder
-(TU Kaiserslautern), and Mohab Safey El Din (Sorbonne University).
+based on Gröbner bases.
 
 This module provide implementations of some operations on polynomial ideals
-based on msolve. Currently the only supported operation is the computation of
-the variety of zero-dimensional ideal over the rationals.
+based on msolve.
 
 Note that msolve must be installed separately.
 
 .. SEEALSO::
 
-- :mod:`sage.features.msolve`
-- :mod:`sage.rings.polynomial.multi_polynomial_ideal`
+    - :mod:`sage.features.msolve`
+    - :mod:`sage.rings.polynomial.multi_polynomial_ideal`
 """
 
 import os
@@ -121,6 +119,18 @@ def variety(ideal, ring, *, proof=True):
     Part of the initial implementation was loosely based on the example
     interfaces available as part of msolve, with the authors' permission.
 
+    EXAMPLES::
+
+        sage: from sage.rings.polynomial.msolve import variety
+        sage: p = 536870909
+        sage: R.<x, y> = PolynomialRing(GF(p), 2, order='lex')
+        sage: I = Ideal([ x*y - 1, (x-2)^2 + (y-1)^2 - 1])
+        sage: sorted(variety(I, GF(p^2), proof=False), key=str) # optional - msolve
+        [{x: 1, y: 1},
+         {x: 254228855*z2 + 114981228, y: 232449571*z2 + 402714189},
+         {x: 267525699, y: 473946006},
+         {x: 282642054*z2 + 154363985, y: 304421338*z2 + 197081624}]
+
     TESTS::
 
         sage: p = 536870909
diff --git a/src/sage/rings/polynomial/multi_polynomial_ideal.py b/src/sage/rings/polynomial/multi_polynomial_ideal.py
index caa74226fd3..243bb9e557b 100644
--- a/src/sage/rings/polynomial/multi_polynomial_ideal.py
+++ b/src/sage/rings/polynomial/multi_polynomial_ideal.py
@@ -4285,8 +4285,8 @@ def groebner_basis(self, algorithm='', deg_bound=None, mult_bound=None, prot=Fal
 
         ALGORITHM:
 
-        Uses Singular, Magma (if available), Macaulay2 (if available),
-        Giac (if available), or a toy implementation.
+        Uses Singular, one of the other systems listed above (if available),
+        or a toy implementation.
 
         TESTS: