cancel returns coeff

sympy/polys/euclidtools.py

@@ -1859,6 +1859,27 @@ def dup_cancel(f, g, K, include=True):
     return dmp_cancel(f, g, 0, K, include=include)
 
 
+def dmp_ZZ_gcd(f, K):
+        from sympy.core.intfunc import igcd
+        assert not f or set([type(i) for i in f]) in [{int},{list}],f
+        def flat(i):
+            if type(i) is list:
+                for i in i:
+                    for j in flat(i):
+                        yield j
+            else:
+                yield i
+        _flat = list(flat(f))
+        if not _flat:
+            return K.one, f
+        if len(_flat) == 1:
+            c = _flat[0]
+        else:
+            c = igcd(*_flat)
+        if c - 1:
+            f = [i//c for i in f] if f and type(f[0]) is int else [[i//c for i in j] for j in f]
+        return c, f
+
 def dmp_cancel(f, g, u, K, include=True):
     """
     Cancel common factors in a rational function `f/g`.
@@ -1881,6 +1902,10 @@ def dmp_cancel(f, g, u, K, include=True):
         cq, f = dmp_clear_denoms(f, u, K0, K, convert=True)
         cp, g = dmp_clear_denoms(g, u, K0, K, convert=True)
         # domain is now K
+    elif K.is_ZZ:
+        # remove gcd
+        cp, f = dmp_ZZ_gcd(f, K)
+        cq, g = dmp_ZZ_gcd(g, K)
     else:
         cp, cq = K.one, K.one
 
@@ -1904,7 +1929,7 @@ def dmp_cancel(f, g, u, K, include=True):
     elif q_neg:
         cp, q = -cp, dmp_neg(q, u, K)
 
-    if include or (c:=(cp,cq)).count(1) == 1 and c.count(-1) == 0:
+    if include:
         p = dmp_mul_ground(p, cp, u, K)
         q = dmp_mul_ground(q, cq, u, K)
         cp = cq = 1

sympy/polys/polytools.py

@@ -7381,7 +7381,13 @@ def cancel(f, *gens, _signsimp=True, **args):
             if not isinstance(f, Tuple):
                 return f.expand()
             else:
-                return S.One, p, q
+                if q:
+                    r = p/q
+                else:
+                    r = p or 1
+                if r < 0:
+                    return -r, S.NegativeOne, S.One
+                return r, S.One, S.One
     except PolynomialError as msg:
         if f.is_commutative and not f.has(Piecewise):
             raise PolynomialError(msg)
@@ -7406,14 +7412,31 @@ def cancel(f, *gens, _signsimp=True, **args):
                     pass
             return f.xreplace(dict(reps))
 
-    c, (P, Q) = 1, F.cancel(G)
+    P, Q = F.cancel(G)
     if opt.get('polys', False) and 'gens' not in opt:
         opt['gens'] = R.symbols
 
     if not isinstance(f, Tuple):
-        return c*(P.as_expr()/Q.as_expr())
+        return P.as_expr()/Q.as_expr()
     else:
         P, Q = P.as_expr(), Q.as_expr()
+        cp, _ = factor_terms(P).as_coeff_Mul()
+        cq, _ = factor_terms(Q).as_coeff_Mul()
+        if cp and cq:
+            c = cp/cq
+            Q = Q/cq
+            P = P/cp
+        elif cp:
+            c = cp
+            P = P/cp
+        elif cq:
+            c = 1/cq
+            Q = Q/cq
+        else:
+            c = S.One
+        if c < 0:
+            P = -P
+            c = -c
         if not opt.get('polys', False):
             return c, P, Q
         else:

sympy/polys/tests/test_euclidtools.py

@@ -662,8 +662,8 @@ def test_dup_cancel():
     p = 2*x + 2
     q = x - 1
 
-    assert R.dup_cancel(f, g) == (p, q)
-    assert R.dup_cancel(f, g, include=False) == (1, 1, p, q)
+    assert R.dup_cancel(f, g) == (p, q),R.dup_cancel(f, g)
+    assert R.dup_cancel(f, g, include=False) == (2, 1, p/2, q)
 
     f = -x - 2
     g = 3*x - 4
@@ -700,7 +700,7 @@ def test_dmp_cancel():
     q = x - 1
 
     assert R.dmp_cancel(f, g) == (p, q)
-    assert R.dmp_cancel(f, g, include=False) == (1, 1, p, q)
+    assert R.dmp_cancel(f, g, include=False) == (2, 1, p/2, q)
 
     assert R.dmp_cancel(0, 0) == (0, 0)
     assert R.dmp_cancel(0, 0, include=False) == (1, 1, 0, 0)

sympy/polys/tests/test_polytools.py

@@ -3301,7 +3301,7 @@ def test_ptcancel():
     raises(ValueError, lambda: cancel((1, 2, 3)))
 
     # tests first tuple return
-    assert (t:=cancel((2, 3))) == (1, 2, 3)
+    assert (t:=cancel((2, 3))) == (S(2)/3, 1, 1)
     assert isinstance(t, tuple)
 
     # tests 2nd tuple return
@@ -3311,11 +3311,11 @@ def test_ptcancel():
 
     # issue 27906
     p, q = Poly(x**2/4 - 1), Poly(x/2 - 1)
-    zz = (1, Poly(x + 2), Poly(2, x))
-    qq = (1, Poly(x + 2, domain=QQ), Poly(2, x, domain=QQ))
+    zz = (S.Half, Poly(x + 2), Poly(1, x))
+    qq = (S.Half, Poly(x + 2, domain=QQ), Poly(1, x, domain=QQ))
     assert p.cancel(q, include=False) == qq
     case1 = p.cancel(q)
-    p, q = [Poly(i) for i in (p/q).as_expr().as_numer_denom()]
+    p, q = [Poly(4*i) for i in (p, q)]
     assert p.cancel(q, include=False) == zz
     case2 = p.cancel(q)
     c,n,d = case1
@@ -3328,15 +3328,15 @@ def test_ptcancel():
     f, g, p, q = 4*x**2 - 4, 2*x - 2, 2*x + 2, 1
     F, G, P, Q = [ Poly(u, x) for u in (f, g, p, q) ]
 
-    assert F.cancel(G) == (1, P, Q)
-    assert cancel((f, g)) == (1, p, q)
-    assert cancel((f, g), x) == (1, p, q)
-    assert cancel((f, g), (x,)) == (1, p, q)
+    assert F.cancel(G) == (2, P/2, Q)
+    assert cancel((f, g)) == (2, p/2, q)
+    assert cancel((f, g), x) == (2, p/2, q)
+    assert cancel((f, g), (x,)) == (2, p/2, q)
     # tests 3rd tuple return
-    assert (t:=cancel((F, G))) == (1, P, Q)
+    assert (t:=cancel((F, G))) == (2, P/2, Q)
     assert isinstance(t, tuple)
-    assert cancel((f, g), polys=True) == (1, P, Q)
-    assert cancel((F, G), polys=False) == (1, p, q)
+    assert cancel((f, g), polys=True) == (2, P/2, Q)
+    assert cancel((F, G), polys=False) == (2, p/2, q)
 
     f = (x**2 - 2)/(x + sqrt(2))
 
@@ -3348,8 +3348,7 @@ def test_ptcancel():
     assert cancel(f) == f
     assert cancel(f, greedy=False) == x + sqrt(2)
 
-    assert cancel((x**2/4 - 1, x/2 - 1)) == (1, x + 2, 2)
-    # assert cancel((x**2/4 - 1, x/2 - 1)) == (S.Half, x + 2, 1)
+    assert cancel((x**2/4 - 1, x/2 - 1)) == (S.Half, x + 2, 1)
 
     assert cancel((x**2 - y)/(x - y)) == 1/(x - y)*(x**2 - y)