sagemathgh-36401: Allow lifting mvpolynomials over non-fields using singular
    
It seems singular now supports this, so pass this functionality on to
users

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URL: sagemath#36401
Reported by: Alex J Best
Reviewer(s): Alex J Best, Kwankyu Lee

src/sage/matrix/matrix0.pyx

@@ -5813,10 +5813,11 @@ cdef class Matrix(sage.structure.element.Matrix):
             Traceback (most recent call last):
             ...
             ArithmeticError: self must be a square matrix
+
             sage: matrix(RR, 1, 1, [2]).inverse_of_unit()                               # needs sage.libs.singular
             Traceback (most recent call last):
             ...
-            NotImplementedError: Lifting of multivariate polynomials over non-fields is not implemented.
+            ArithmeticError: non-invertible matrix
 
             sage: R = ZZ.cartesian_product(ZZ)
             sage: m = matrix(R, 2, [R((2,1)), R((1,1)), R((1,1)), R((1,2))])

src/sage/rings/polynomial/multi_polynomial_libsingular.pyx

@@ -4490,8 +4490,25 @@ cdef class MPolynomial_libsingular(MPolynomial_libsingular_base):
             Traceback (most recent call last):
             ...
             ValueError: polynomial is not in the ideal
+
+        Check that we can work over the integers::
+
+            sage: R.<x1,x2> = ZZ[]
+            sage: I = R.ideal(x2**2 + x1 - 2, x1**2 - 1)
+            sage: f = I.gen(0) + x2*I.gen(1)
             sage: f.lift(I)
             [1, x2]
+            sage: (f+1).lift(I)
+            Traceback (most recent call last):
+            ...
+            ValueError: polynomial is not in the ideal
+            sage: A.<x,y> = PolynomialRing(ZZ,2,order='degrevlex')
+            sage: I = A.ideal([x^10 + x^9*y^2, y^8 - x^2*y^7 ])
+            sage: f = x*y^13 + y^12
+            sage: M = f.lift(I)
+            sage: M
+            [y^7, x^7*y^2 + x^8 + x^5*y^3 + x^6*y + x^3*y^4 + x^4*y^2 + x*y^5 + x^2*y^3 + y^4]
+
 
         TESTS:
 
@@ -4509,8 +4526,6 @@ cdef class MPolynomial_libsingular(MPolynomial_libsingular_base):
 
         """
         global errorreported
-        if not self._parent._base.is_field():
-            raise NotImplementedError("Lifting of multivariate polynomials over non-fields is not implemented.")
 
         cdef ideal *fI = idInit(1,1)
         cdef ideal *_I