gh-35021:  Implement check for Lorentzian polynomials #28252 
    
Fixes #28252

### 📚 Description

<!-- Describe your changes in detail -->
<!-- Why is this change required? What problem does it solve? -->
<!-- If it resolves an open issue, please link to the issue here. For
example "Closes #1337" -->

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->
<!-- If your change requires a documentation PR, please link it
appropriately -->
<!-- If you're unsure about any of these, don't hesitate to ask. We're
here to help! -->

- [ ] I have linked an issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation accordingly.

### ⌛ Dependencies
<!-- List all open pull requests that this PR logically depends on -->
<!--
- #xyz: short description why this is a dependency
- #abc: ...
-->
    
URL: #35021
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

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

@@ -803,6 +803,10 @@ REFERENCES:
              Stein. strassen_window_multiply_c. strassen.pyx, Sage
              3.0, 2008. http://www.sagemath.org
 
+.. [BrHu2019] Petter Brändén, June Huh. *Lorentzian polynomials*.
+              Ann. Math. (2) 192, No. 3, 821-891 (2020).
+              :arxiv:`1902.03719`, :doi:`10.4007/annals.2020.192.3.4`.
+
 .. [BHNR2004] \S. Brlek, S. Hamel, M. Nivat, C. Reutenauer, On the
               Palindromic Complexity of Infinite Words,
               in J. Berstel, J.  Karhumaki, D. Perrin, Eds,
@@ -3247,6 +3251,11 @@ REFERENCES:
             Designs, Codes and Cryptography 8 (1996) 145-157.
             :doi:`10.1023/A:1018037025910`.
 
+.. [HMMS2019] June Huh, Jacob P. Matherne, Karola Mészáros, Avery St. Dizier.
+              *Logarithmic concavity of Schur and related polynomials*.
+              Trans. Am. Math. Soc. 375, No. 6, 4411-4427 (2022).
+              :arxiv:`1906.09633`, :doi:`10.1090/tran/8606`.
+
 .. [Hutz2007] \B. Hutz. Arithmetic Dynamics on Varieties of dimension greater
               than one. PhD Thesis, Brown University 2007
 

src/sage/quadratic_forms/quadratic_form.py

@@ -30,13 +30,15 @@
 from sage.arith.all import GCD, LCM
 from sage.rings.all import Ideal, QQ
 from sage.rings.ring import is_Ring, PrincipalIdealDomain
-from sage.structure.sage_object import SageObject
 from sage.structure.element import is_Vector
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+from sage.rings.polynomial.polynomial_element import is_Polynomial
+from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial
 from sage.modules.free_module_element import vector
 from sage.quadratic_forms.genera.genus import genera
 from sage.quadratic_forms.quadratic_form__evaluate import QFEvaluateVector, QFEvaluateMatrix
-
+from sage.structure.sage_object import SageObject
+from sage.combinat.integer_lists.invlex import IntegerListsLex
 
 def QuadraticForm__constructor(R, n=None, entries=None):
     """
@@ -206,6 +208,10 @@ class QuadraticForm(SageObject):
          in `R` (given lexicographically, or equivalently, by rows of the
          matrix)
 
+    #. ``QuadraticForm(p)``, where
+
+      - `p` -- a homogeneous polynomial of degree `2`
+
     #. ``QuadraticForm(R, n)``, where
 
        - `R` -- a ring
@@ -284,6 +290,16 @@ class QuadraticForm(SageObject):
         [ 1 5 ]
         [ * 4 ]
 
+    ::
+
+        sage: P.<x,y,z> = QQ[]
+        sage: p = x^2 + 2*x*y + x*z/2 + y^2 + y*z/3
+        sage: QuadraticForm(p)
+        Quadratic form in 3 variables over Rational Field with coefficients:
+        [ 1 2 1/2 ]
+        [ * 1 1/3 ]
+        [ * * 0 ]
+
     ::
 
         sage: QuadraticForm(ZZ, m + m.transpose())
@@ -499,6 +515,25 @@ def __init__(self, R, n=None, entries=None, unsafe_initialization=False, number_
             sage: s.dim()
             4
 
+            sage: P.<x,y,z> = QQ[]
+            sage: p = x^2 + y^2 + 2*x*z
+            sage: QuadraticForm(p)
+            Quadratic form in 3 variables over Rational Field with coefficients:
+            [ 1 0 2 ]
+            [ * 1 0 ]
+            [ * * 0 ]
+            sage: z = P.zero()
+            sage: QuadraticForm(z)
+            Quadratic form in 3 variables over Rational Field with coefficients:
+            [ 0 0 0 ]
+            [ * 0 0 ]
+            [ * * 0 ]
+            sage: q = x^2 + 3*y - z
+            sage: QuadraticForm(q)
+            Traceback (most recent call last):
+            ...
+            ValueError: polynomial is neither zero nor homogeneous of degree 2
+
         TESTS::
 
             sage: s == loads(dumps(s))
@@ -510,6 +545,10 @@ def __init__(self, R, n=None, entries=None, unsafe_initialization=False, number_
 
             sage: x = polygen(ZZ, 'x')
             sage: QuadraticForm(x**2)
+            Quadratic form in 1 variables over Integer Ring with coefficients:
+            [ 1 ]
+
+            sage: QuadraticForm(1)
             Traceback (most recent call last):
             ....
             TypeError: wrong input for QuadraticForm
@@ -527,20 +566,39 @@ def __init__(self, R, n=None, entries=None, unsafe_initialization=False, number_
                 M_ring = R
                 matrix_init_flag = True
 
-        elif not is_Matrix(R):
-            # first argument, if not a ring, must be a matrix
-            raise TypeError('wrong input for QuadraticForm')
-        else:
-            # Deal with:  QuadraticForm(matrix)
+        elif is_Matrix(R):
+            M = R
+
             # Test if R is symmetric and has even diagonal
-            if not self._is_even_symmetric_matrix_(R):
+            if not self._is_even_symmetric_matrix_(M):
                 raise TypeError("the matrix is not a symmetric with even diagonal")
 
-            # Rename the matrix and ring
-            M = R
-            M_ring = R.base_ring()
+            M_ring = M.base_ring()
             matrix_init_flag = True
 
+        elif is_Polynomial(R) or is_MPolynomial(R):
+            p = R
+            
+            if not p.is_zero() and not (p.is_homogeneous() and p.degree() == 2):
+                raise ValueError("polynomial is neither zero nor homogeneous of degree 2")
+
+            P = p.parent()
+            R, n = P.base_ring(), P.ngens()
+
+            # Extract quadratic form coefficients
+            entries = []
+            if n == 0:
+                exponents = []
+            elif n == 1:
+                exponents = [2]
+            else:
+                exponents = IntegerListsLex(2, length=n)
+            for alpha in exponents:
+                entries.append(p[alpha])
+
+        else:
+            raise TypeError('wrong input for QuadraticForm')
+
         # Perform the quadratic form initialization
         if matrix_init_flag:
             self.__n = ZZ(M.nrows())