added polynomial evaluation at algebraic point

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

src/api/python/z3.py

@@ -1126,6 +1126,27 @@ def Var(idx, s):
         _z3_assert(is_sort(s), "Z3 sort expected")
     return _to_expr_ref(Z3_mk_bound(s.ctx_ref(), idx, s.ast), s.ctx)
 
+def RealVar(idx, ctx=None):
+    """
+    Create a real free variable. Free variables are used to create quantified formulas.
+    They are also used to create polynomials.
+    
+    >>> RealVar(0)
+    Var(0)
+    """
+    return Var(idx, RealSort(ctx))
+
+def RealVarVector(n, ctx=None):
+    """
+    Create a list of Real free variables.
+    The variables have ids: 0, 1, ..., n-1
+    
+    >>> x0, x1, x2, x3 = RealVarVector(4)
+    >>> x2
+    Var(2)
+    """
+    return [ RealVar(i, ctx) for i in range(n) ]
+
 #########################################
 #
 # Booleans

src/api/python/z3num.py

@@ -481,6 +481,30 @@ class Numeral:
 
     def ctx_ref(self):
         return self.ctx.ref()
+
+def eval_sign_at(p, vs):
+    """ 
+    Evaluate the sign of the polynomial `p` at `vs`.  `p` is a Z3
+    Expression containing arithmetic operators: +, -, *, ^k where k is
+    an integer; and free variables x that is_var(x) is True. Moreover,
+    all variables must be real.
+    
+    The result is 1 if the polynomial is positive at the given point,
+    -1 if negative, and 0 if zero.
+
+    >>> x0, x1, x2 = RealVarVector(3)
+    >>> eval_sign_at(x0**2 + x1*x2 + 1, (Numeral(0), Numeral(1), Numeral(2)))
+    1
+    >>> eval_sign_at(x0**2 - 2, [ Numeral(Sqrt(2)) ])
+    0
+    >>> eval_sign_at((x0 + x1)*(x0 + x2), (Numeral(0), Numeral(Sqrt(2)), Numeral(Sqrt(3))))
+    1
+    """
+    num = len(vs)
+    _vs = (Ast * num)()
+    for i in range(num):
+        _vs[i] = vs[i].ast
+    return Z3_algebraic_eval(p.ctx_ref(), p.as_ast(), num, _vs)
         
 if __name__ == "__main__":
     import doctest