gh-35174: cleaning and enhancement to PolyDict

    
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### 📚 Description

Fixes #34000 (and more complete version of the proposition at #35020).

A discussion about the general problem of inexact zeros has been opened
at #35319

### 📝 Checklist

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URL: #35174
Reported by: Vincent Delecroix
Reviewer(s): Frédéric Chapoton, Travis Scrimshaw, Vincent Delecroix

src/sage/rings/polynomial/hilbert.pyx

@@ -133,9 +133,8 @@ cdef list quotient_by_var(list L, size_t index):
     cdef list result = L[:len(L)] # creates a copy
     cdef size_t i
     for i in range(len(L)):
-        m_j = (<ETuple>PyList_GET_ITEM(L,i)).divide_by_var(index)
-        if m_j is not None:
-            result.append(m_j)
+        if (<ETuple> PyList_GET_ITEM(L, i)).get_exp(index):
+            result.append((<ETuple> PyList_GET_ITEM(L, i)).divide_by_var(index))
     return interred(result)
 
 cdef ETuple sum_from_list(list L, size_t s, size_t l):

src/sage/rings/polynomial/laurent_polynomial.pyx

@@ -12,9 +12,10 @@ Elements of Laurent polynomial rings
 
 from sage.rings.integer cimport Integer
 from sage.categories.map cimport Map
-from sage.structure.element import is_Element, coerce_binop
+from sage.structure.element import is_Element, coerce_binop, parent
 from sage.structure.factorization import Factorization
 from sage.misc.derivative import multi_derivative
+from sage.rings.polynomial.polydict cimport monomial_exponent
 from sage.rings.polynomial.polynomial_element import Polynomial
 from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
 from sage.structure.richcmp cimport richcmp, rich_to_bool
@@ -2254,12 +2255,12 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial):
         cdef dict D = <dict> self._poly.dict()
         cdef dict DD
         if self._mon.is_constant():
-            self._prod = PolyDict(D, force_etuples=False)
+            self._prod = PolyDict(D)
             return
         DD = {}
         for k in D:
             DD[k.eadd(self._mon)] = D[k]
-        self._prod = PolyDict(DD, force_etuples=False)
+        self._prod = PolyDict(DD)
 
     def is_unit(self):
         """
@@ -2556,19 +2557,18 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial):
             sage: f.monomial_coefficient(x + y)
             Traceback (most recent call last):
             ...
-            ValueError: input must be a monomial
+            ValueError: not a monomial
         """
-        if mon.parent() != self._parent:
+        if parent(mon) != self._parent:
             raise TypeError("input must have the same parent")
         cdef LaurentPolynomial_mpair m = <LaurentPolynomial_mpair> mon
         if m._prod is None:
             m._compute_polydict()
-        if len(m._prod) != 1:
-            raise ValueError("input must be a monomial")
         if self._prod is None:
             self._compute_polydict()
-        c = self._prod.monomial_coefficient(m._prod.dict())
-        return self._parent.base_ring()(c)
+        exp = monomial_exponent(m._prod)
+        zero = self._parent.base_ring().zero()
+        return self._prod.get(exp, zero)
 
     def constant_coefficient(self):
         """

src/sage/rings/polynomial/multi_polynomial.pyx

@@ -2347,7 +2347,7 @@ cdef class MPolynomial(CommutativePolynomial):
             sage: F.reduced_form(prec=50, smallest_coeffs=False)
             Traceback (most recent call last):
             ...
-            ValueError: accuracy of Newton's root not within tolerance(0.0000124... > 1e-06), increase precision
+            ValueError: accuracy of Newton's root not within tolerance(0.000012... > 1e-06), increase precision
             sage: F.reduced_form(prec=100, smallest_coeffs=False)
             (
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