Make all polynomials handle negative powers

src/flint/flint_base/flint_base.pyx

@@ -6,6 +6,7 @@ from flint.flintlib.types.flint cimport (
 )
 from flint.utils.flint_exceptions import DomainError
 from flint.flintlib.types.mpoly cimport ordering_t
+from flint.flintlib.functions.fmpz cimport fmpz_cmp_si
 from flint.flint_base.flint_context cimport thectx
 from flint.utils.typecheck cimport typecheck
 cimport libc.stdlib
@@ -725,14 +726,15 @@ cdef class flint_mpoly(flint_elem):
     def __pow__(self, other, modulus):
         if modulus is not None:
             raise NotImplementedError("cannot specify modulus outside of the context")
-        elif typecheck(other, fmpz):
-            return self._pow_(other)
 
-        other = any_as_fmpz(other)
-        if other is NotImplemented:
-            return NotImplemented
-        elif other < 0:
-            raise ValueError("cannot raise to a negative power")
+        if not typecheck(other, fmpz):
+            other = any_as_fmpz(other)
+            if other is NotImplemented:
+                return NotImplemented
+
+        if fmpz_cmp_si((<fmpz>other).val, 0) < 0:
+            self = 1 / self
+            other = -other
 
         return self._pow_(other)
 

src/flint/test/test_all.py

@@ -402,7 +402,7 @@ def test_fmpz_poly():
     assert raises(lambda: [] // Z([1,2]), TypeError)
     assert raises(lambda: [] % Z([1,2]), TypeError)
     assert raises(lambda: divmod([], Z([1,2])), TypeError)
-    assert raises(lambda: Z([1,2,3]) ** -1, (OverflowError, ValueError))
+    assert raises(lambda: Z([1,2,3]) ** -1, DomainError)
     assert raises(lambda: Z([1,2,3]) ** Z([1,2]), TypeError)
     assert raises(lambda: Z([1,2]) // Z([]), ZeroDivisionError)
     assert raises(lambda: Z([]) // Z([]), ZeroDivisionError)
@@ -2109,7 +2109,7 @@ def test_fmpz_mod_poly():
         assert (f + 1) // f == 1
 
         # pow
-        assert raises(lambda: f**(-2), ValueError)
+        assert raises(lambda: f**(-2), DomainError)
         assert f*f == f**2
         assert f*f == f**fmpz(2)
 
@@ -2768,7 +2768,7 @@ def setbad(obj, i, val):
         assert P([1, 1]) ** 0 == P([1])
         assert P([1, 1]) ** 1 == P([1, 1])
         assert P([1, 1]) ** 2 == P([1, 2, 1])
-        assert raises(lambda: P([1, 1]) ** -1, ValueError)
+        assert raises(lambda: P([1, 1]) ** -1, DomainError)
         assert raises(lambda: P([1, 1]) ** None, TypeError)
 
         # XXX: Not sure what this should do in general:
@@ -3254,7 +3254,7 @@ def quick_poly():
             (0, 1): 4,
             (0, 0): 1,
         })
-        assert raises(lambda: P(ctx=ctx) ** -1, ValueError)
+        assert raises(lambda: P(ctx=ctx) ** -1, ZeroDivisionError)
         assert raises(lambda: P(ctx=ctx) ** None, TypeError)
 
         # # XXX: Not sure what this should do in general:
@@ -3464,6 +3464,32 @@ def _all_polys_mpolys():
         yield P, S, [x, y], is_field, characteristic
 
 
+def test_properties_poly_mpoly():
+    """Test is_zero, is_one etc for all polynomials."""
+    for P, S, [x, y], is_field, characteristic in _all_polys_mpolys():
+
+        zero = 0*x
+        one = zero + 1
+        two = one + 1
+
+        assert zero.is_zero() is True
+        assert one.is_zero() is False
+        assert two.is_zero() is False
+        assert x.is_zero() is False
+
+        assert zero.is_one() is False
+        assert one.is_one() is True
+        assert two.is_one() is False
+        assert x.is_one() is False
+
+        assert zero.is_constant() is True
+        assert one.is_constant() is True
+        assert two.is_constant() is True
+        assert x.is_constant() is False
+
+        # is_gen?
+
+
 def test_factor_poly_mpoly():
     """Test that factor() is consistent across different poly/mpoly types."""
 
@@ -3671,6 +3697,52 @@ def factor_sqf(p):
                 assert (2*(x+y)).gcd(4*(x+y)**2) == x + y
 
 
+def test_division_poly_mpoly():
+    """Test that division is consistent across different poly/mpoly types."""
+
+    Z = flint.fmpz
+
+    for P, S, [x, y], is_field, characteristic in _all_polys_mpolys():
+
+        if characteristic != 0 and not characteristic.is_prime():
+            # nmod_poly crashes for many operations with non-prime modulus
+            #     https://github.com/flintlib/python-flint/issues/124
+            # so we can't even test it...
+            nmod_poly_will_crash = type(x) is flint.nmod_poly
+            if nmod_poly_will_crash:
+                continue
+
+        one = x**0 # 1 as a polynomial
+        two = one + one
+
+        if is_field or characteristic == 0:
+            assert x / x == x**0 == 1 == one
+            assert x / 1 == x / S(1) == x / one == x**1 == x
+            assert 1 / one == one**-1 == one**Z(-1) == 1, type(one)
+            assert -1 / one == 1 / -one == (-one)**-1 == (-one)**Z(-1) == -one == -1
+            assert (-one) ** -2 == (-one)**Z(-2) == one
+            assert raises(lambda: 1 / x, DomainError)
+            assert raises(lambda: x ** -1, DomainError)
+
+        if is_field:
+            half = S(1)/2 * one # 1/2 as a polynomial
+            assert half == S(1)/2
+            assert x / half == 2*x
+            assert 1 / half == S(1) / half == one / half == one / (S(1)/2) == 2
+            assert half ** -1 == half ** Z(-1) == 2
+            assert two ** -1 == two ** Z(-1) == half
+        elif characteristic == 0:
+            assert raises(lambda: x / 2, DomainError)
+            assert raises(lambda: x / two, DomainError), characteristic
+            assert raises(lambda: two ** -1, DomainError)
+            assert raises(lambda: two ** Z(-1), DomainError)
+        else:
+            # Non-prime modulus...
+            # nmod can crash and fmpz_mod_poly won't crash but has awkward
+            # behaviour under division.
+            pass
+
+
 def _all_matrices():
     """Return a list of matrix types and scalar types."""
     R163 = flint.fmpz_mod_ctx(163)
@@ -4569,7 +4641,7 @@ def test_fq_default_poly():
         # pow
         # assert ui and fmpz exp agree for polynomials and generators
         R_gen = R_test.gen()
-        assert raises(lambda: f**(-2), ValueError)
+        assert raises(lambda: f**(-2), DomainError)
         assert pow(f, 2**60, g) == pow(pow(f, 2**30, g), 2**30, g)
         assert pow(R_gen, 2**60, g) == pow(pow(R_gen, 2**30, g), 2**30, g)
         assert raises(lambda: pow(f, -2, g), ValueError)
@@ -4698,7 +4770,9 @@ def test_all_tests():
     test_division_poly,
     test_division_matrix,
 
+    test_properties_poly_mpoly,
     test_factor_poly_mpoly,
+    test_division_poly_mpoly,
 
     test_polys,
     test_mpolys,

src/flint/types/fmpq_mpoly.pyx

@@ -275,6 +275,19 @@ cdef class fmpq_mpoly(flint_mpoly):
     def is_one(self):
         return <bint>fmpq_mpoly_is_one(self.val, self.ctx.val)
 
+    def is_constant(self):
+        """
+        Returns True if this is a constant polynomial.
+
+        >>> R = fmpq_mpoly_ctx.get(['x', 'y'])
+        >>> x, y = R.gens()
+        >>> x.is_constant()
+        False
+        >>> (0*x + 1).is_constant()
+        True
+        """
+        return self.total_degree() <= 0
+
     def __richcmp__(self, other, int op):
         if not (op == Py_EQ or op == Py_NE):
             return NotImplemented

src/flint/types/fmpq_poly.pyx

@@ -171,11 +171,29 @@ cdef class fmpq_poly(flint_poly):
         return not fmpq_poly_is_zero(self.val)
 
     def is_zero(self):
+        """
+        Returns True if this is the zero polynomial.
+        """
         return <bint>fmpq_poly_is_zero(self.val)
 
     def is_one(self):
+        """
+        Returns True if this polynomial is equal to 1.
+        """
         return <bint>fmpq_poly_is_one(self.val)
 
+    def is_constant(self):
+        """
+        Returns True if this polynomial is a scalar (constant).
+
+        >>> f = fmpq_poly([0, 1])
+        >>> f
+        x
+        >>> f.is_constant()
+        False
+        """
+        return fmpq_poly_degree(self.val) <= 0
+
     def leading_coefficient(self):
         """
         Returns the leading coefficient of the polynomial.
@@ -374,7 +392,8 @@ cdef class fmpq_poly(flint_poly):
         if mod is not None:
             raise NotImplementedError("fmpz_poly modular exponentiation")
         if exp < 0:
-            raise ValueError("fmpq_poly negative exponent")
+            self = 1 / self
+            exp = -exp
         res = fmpq_poly.__new__(fmpq_poly)
         fmpq_poly_pow(res.val, self.val, <ulong>exp)
         return res

src/flint/types/fmpz_mod_mpoly.pyx

@@ -296,6 +296,19 @@ cdef class fmpz_mod_mpoly(flint_mpoly):
     def is_one(self):
         return <bint>fmpz_mod_mpoly_is_one(self.val, self.ctx.val)
 
+    def is_constant(self):
+        """
+        Returns True if this is a constant polynomial.
+
+        >>> R = fmpz_mod_mpoly_ctx.get(['x', 'y'], modulus=11)
+        >>> x, y = R.gens()
+        >>> x.is_constant()
+        False
+        >>> (0*x + 1).is_constant()
+        True
+        """
+        return self.total_degree() <= 0
+
     def __richcmp__(self, other, int op):
         if not (op == Py_EQ or op == Py_NE):
             return NotImplemented

src/flint/types/fmpz_mod_poly.pyx

@@ -544,7 +544,8 @@ cdef class fmpz_mod_poly(flint_poly):
 
         cdef fmpz_mod_poly res
         if e < 0:
-            raise ValueError("Exponent must be non-negative")
+            self = 1 / self
+            e = -e
 
         cdef ulong e_ulong = e
         res = self.ctx.new_ctype_poly()

src/flint/types/fmpz_mpoly.pyx

@@ -261,6 +261,20 @@ cdef class fmpz_mpoly(flint_mpoly):
     def is_one(self):
         return <bint>fmpz_mpoly_is_one(self.val, self.ctx.val)
 
+    def is_constant(self):
+        """
+        Returns True if this is a constant polynomial.
+
+        >>> ctx = fmpz_mpoly_ctx.get(['x', 'y'])
+        >>> x, y = ctx.gens()
+        >>> p = x**2 + y
+        >>> p.is_constant()
+        False
+        >>> (0*p + 1).is_constant()
+        True
+        """
+        return self.total_degree() <= 0
+
     def __richcmp__(self, other, int op):
         if not (op == Py_EQ or op == Py_NE):
             return NotImplemented

src/flint/types/fmpz_poly.pyx

@@ -141,11 +141,36 @@ cdef class fmpz_poly(flint_poly):
         return not fmpz_poly_is_zero(self.val)
 
     def is_zero(self):
+        """
+        True if this polynomial is the zero polynomial.
+
+        >>> fmpz_poly([]).is_zero()
+        True
+        """
         return <bint>fmpz_poly_is_zero(self.val)
 
     def is_one(self):
+        """
+        True if this polynomial is equal to one.
+
+        >>> fmpz_poly([2]).is_one()
+        False
+        """
         return <bint>fmpz_poly_is_one(self.val)
 
+    def is_constant(self):
+        """
+        True if this is a constant polynomial.
+
+        >>> x = fmpz_poly([0, 1])
+        >>> two = fmpz_poly([2])
+        >>> x.is_constant()
+        False
+        >>> two.is_constant()
+        True
+        """
+        return fmpz_poly_degree(self.val) <= 0
+
     def leading_coefficient(self):
         """
         Returns the leading coefficient of the polynomial.
@@ -348,7 +373,9 @@ cdef class fmpz_poly(flint_poly):
         if mod is not None:
             raise NotImplementedError("fmpz_poly modular exponentiation")
         if exp < 0:
-            raise ValueError("fmpz_poly negative exponent")
+            if not fmpz_poly_is_unit(self.val):
+                raise DomainError("fmpz_poly negative exponent, non-unit base")
+            exp = -exp
         res = fmpz_poly.__new__(fmpz_poly)
         fmpz_poly_pow(res.val, self.val, <ulong>exp)
         return res

src/flint/types/fq_default_poly.pyx

@@ -642,7 +642,8 @@ cdef class fq_default_poly(flint_poly):
 
         cdef fq_default_poly res
         if e < 0:
-            raise ValueError("Exponent must be non-negative")
+            self = 1 / self
+            e = -e
 
         if e == 2:
             return self.square()

src/flint/types/nmod_mpoly.pyx

@@ -275,6 +275,12 @@ cdef class nmod_mpoly(flint_mpoly):
     def is_one(self):
         return <bint>nmod_mpoly_is_one(self.val, self.ctx.val)
 
+    def is_constant(self):
+        """
+        Returns True if this is a constant polynomial.
+        """
+        return self.total_degree() <= 0
+
     def __richcmp__(self, other, int op):
         if not (op == Py_EQ or op == Py_NE):
             return NotImplemented
@@ -294,7 +300,8 @@ cdef class nmod_mpoly(flint_mpoly):
                 self.ctx.val
             )
         elif isinstance(other, int):
-            return (op == Py_NE) ^ <bint>nmod_mpoly_equal_ui(self.val, other, self.ctx.val)
+            omod = other % self.ctx.modulus()
+            return (op == Py_NE) ^ <bint>nmod_mpoly_equal_ui(self.val, omod, self.ctx.val)
         else:
             return NotImplemented