From 3a649e4b39172dc29ad79a54fa8f1f9284a0cd21 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Fr=C3=A9d=C3=A9ric=20Chapoton?= <chapoton@unistra.fr>
Date: Sat, 13 Sep 2025 20:16:40 +0200
Subject: [PATCH] add Newton polytopes for Laurent polynomials

---
 .../rings/polynomial/laurent_polynomial.pyx   | 24 +++++++++++++++++++
 .../polynomial/laurent_polynomial_mpair.pyx   | 23 ++++++++++++++++++
 2 files changed, 47 insertions(+)

diff --git a/src/sage/rings/polynomial/laurent_polynomial.pyx b/src/sage/rings/polynomial/laurent_polynomial.pyx
index 81c110ebd94..d17cb3cc630 100644
--- a/src/sage/rings/polynomial/laurent_polynomial.pyx
+++ b/src/sage/rings/polynomial/laurent_polynomial.pyx
@@ -913,6 +913,30 @@ cdef class LaurentPolynomial_univariate(LaurentPolynomial):
         """
         return [i + self.__n for i in self.__u.exponents()]
 
+    def newton_polytope(self):
+        r"""
+        Return the Newton polytope of this Laurent polynomial.
+
+        EXAMPLES::
+
+            sage: R.<x> = LaurentPolynomialRing(QQ)
+            sage: f = 1 + x + 33 * x^-3
+            sage: P = f.newton_polytope(); P                                            # needs sage.geometry.polyhedron
+            A 1-dimensional polyhedron in ZZ^1 defined as the convex hull of 2 vertices
+
+        TESTS::
+
+            sage: R.<x> = LaurentPolynomialRing(QQ)
+            sage: R(0).newton_polytope()                                                # needs sage.geometry.polyhedron
+            The empty polyhedron in ZZ^0
+            sage: R(1).newton_polytope()                                                # needs sage.geometry.polyhedron
+            A 0-dimensional polyhedron in ZZ^1 defined as the convex hull of 1 vertex
+        """
+        from sage.geometry.polyhedron.constructor import Polyhedron
+        from sage.rings.integer_ring import ZZ
+        return Polyhedron(vertices=[(e,) for e in self.exponents()],
+                          base_ring=ZZ)
+
     def __setitem__(self, n, value):
         """
         EXAMPLES::
diff --git a/src/sage/rings/polynomial/laurent_polynomial_mpair.pyx b/src/sage/rings/polynomial/laurent_polynomial_mpair.pyx
index 67446b49e25..6aa37df3611 100644
--- a/src/sage/rings/polynomial/laurent_polynomial_mpair.pyx
+++ b/src/sage/rings/polynomial/laurent_polynomial_mpair.pyx
@@ -1325,6 +1325,29 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial):
         # Find the minimal valuation of x by checking each term
         return Integer(min(e[i] for e in self.exponents()))
 
+    def newton_polytope(self):
+        r"""
+        Return the Newton polytope of this Laurent polynomial.
+
+        EXAMPLES::
+
+            sage: R.<x, y> = LaurentPolynomialRing(QQ)
+            sage: f = 1 + x*y + y**2 + 33 * x^-3
+            sage: P = f.newton_polytope(); P                                            # needs sage.geometry.polyhedron
+            A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices
+
+        TESTS::
+
+            sage: R.<x,y> = LaurentPolynomialRing(QQ)
+            sage: R(0).newton_polytope()                                                # needs sage.geometry.polyhedron
+            The empty polyhedron in ZZ^0
+            sage: R(1).newton_polytope()                                                # needs sage.geometry.polyhedron
+            A 0-dimensional polyhedron in ZZ^2 defined as the convex hull of 1 vertex
+        """
+        from sage.geometry.polyhedron.constructor import Polyhedron
+        from sage.rings.integer_ring import ZZ
+        return Polyhedron(vertices=self.exponents(), base_ring=ZZ)
+
     def has_inverse_of(self, i):
         """
         INPUT: