change polynomial_construction from the review, update description and add doctests

Author: enriqueartal

src/sage/rings/polynomial/laurent_polynomial_mpair.pyx

@@ -1522,8 +1522,7 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial):
 
     def polynomial_construction(self):
         """
-        Return a polynomial having no monomial as a factor
-        and the shift monomial.
+        Factor ``self`` into a polynomial and a monomial.
 
         OUTPUT:
 
@@ -1535,16 +1534,27 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial):
             sage: R.<x, y> = LaurentPolynomialRing(QQ)
             sage: f = y / x + x^2 / y + 3 * x^4 * y^-2
             sage: f.polynomial_construction()
-            (3*x^5 + x^3*y + y^3, x^-1*y^-2)
+            (3*x^5 + x^3*y + y^3, 1/(x*y^2))
+            sage: f = y * x + x^2 / y + 3 * x^4 * y^-2
+            sage: f.polynomial_construction()
+             (3*x^3 + y^3 + x*y, x/y^2)
+            sage: x.polynomial_construction()
+            (1, x)
+            sage: (y^-1).polynomial_construction()
+            (1, 1/y)
         """
-        R = self.parent()
-        n = R.ngens()
-        S = R.polynomial_ring()
-        if not self:
-            return (self, R(1))
-        minimo = tuple(min(a[1].degree(v) for a in self) for v in R.gens())
-        mon = R({minimo: 1})
-        return (S(mon ** -1 * self), mon)
+        self._normalize()
+        ring = self._parent._R
+        g = ring.gens()
+        mon = ring.prod(g[i] ** j for i, j in enumerate(self._mon))
+        return (self._poly, mon)        # R = self.parent()
+        # n = R.ngens()
+        # S = R.polynomial_ring()
+        # if not self:
+        #     return (self, R(1))
+        # minimo = tuple(min(a[1].degree(v) for a in self) for v in R.gens())
+        # mon = R({minimo: 1})
+        # return (S(mon ** -1 * self), mon)
 
     def factor(self):
         """