# HG changeset patch # User Nick Alexander # Date 1241902740 25200 # Branch symbolics_switch # Node ID d4f2118fb20e3705214e799a35b9bb61799d4814 # Parent 2d4bc47a082f33c74612a0d9a0589901c78ae138 [mq]: elliptic_curves.patch diff -r 2d4bc47a082f -r d4f2118fb20e sage/schemes/elliptic_curves/ell_generic.py --- a/sage/schemes/elliptic_curves/ell_generic.py Sat May 09 11:09:27 2009 -0700 +++ b/sage/schemes/elliptic_curves/ell_generic.py Sat May 09 13:59:00 2009 -0700 @@ -359,9 +359,6 @@ sage: eqn = symbolic_expression(E); eqn y^2 + y == x^3 - x^2 - 10*x - 20 - sage: print eqn - 2 3 2 - y + y == x - x - 10 x - 20 We verify that the given point is on the curve:: @@ -372,58 +369,60 @@ We create a single expression:: - sage: F = eqn.lhs() - eqn.rhs(); print F - 2 3 2 - y + y - x + x + 10 x + 20 + sage: F = eqn.lhs() - eqn.rhs(); F + -x^3 + x^2 + y^2 + 10*x + y + 20 sage: y = var('y') - sage: print F.solve(y) - [ - 3 2 - - sqrt(4 x - 4 x - 40 x - 79) - 1 - y == ----------------------------------- - 2, - 3 2 - sqrt(4 x - 4 x - 40 x - 79) - 1 - y == --------------------------------- - 2 - ] + sage: F.solve(y) + [y == -1/2*sqrt(4*x^3 - 4*x^2 - 40*x - 79) - 1/2, + y == 1/2*sqrt(4*x^3 - 4*x^2 - 40*x - 79) - 1/2] You can also solve for x in terms of y, but the result is horrendous. Continuing with the above example, we can explicitly find points over random fields by substituting in values for x:: - sage: v = F.solve(y)[0].rhs() - sage: print v - 3 2 - - sqrt(4 x - 4 x - 40 x - 79) - 1 - ----------------------------------- - 2 + sage: v = F.solve(y)[0].rhs(); v + -1/2*sqrt(4*x^3 - 4*x^2 - 40*x - 79) - 1/2 sage: v = v.function(x) sage: v(3) - (-sqrt(127)*I - 1)/2 + -1/2*sqrt(-127) - 1/2 sage: v(7) - (-sqrt(817) - 1)/2 + -1/2*sqrt(817) - 1/2 sage: v(-7) - (-sqrt(1367)*I - 1)/2 + -1/2*sqrt(-1367) - 1/2 sage: v(sqrt(2)) - (-sqrt(-32*sqrt(2) - 87) - 1)/2 + -1/2*sqrt(-32*sqrt(2) - 87) - 1/2 We can even do arithmetic with them, as follows:: sage: E2 = E.change_ring(SR); E2 - Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Symbolic Ring - sage: P = E2.point((3, v(3), 1), check=False) + Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Symbolic Ring + sage: P = E2.point((3, v(3), 1), check=False) # the check=False option doesn't verify that y^2 = f(x) sage: P - (3 : (-sqrt(127)*I - 1)/2 : 1) + (3 : -1/2*sqrt(-127) - 1/2 : 1) sage: P + P - (-756/127 : (sqrt(127)*I + 1)/2 + 12507*I/(127*sqrt(127)) - 1 : 1) + (-756/127 : 41143/32258*sqrt(-127) - 1/2 : 1) We can even throw in a transcendental:: sage: w = E2.point((pi,v(pi),1), check=False); w - (pi : (-sqrt(4*pi^3 - 4*pi^2 - 40*pi - 79) - 1)/2 : 1) + (pi : -1/2*sqrt(-40*pi + 4*pi^3 - 4*pi^2 - 79) - 1/2 : 1) + sage: x, y, z = w; ((y^2 + y) - (x^3 - x^2 - 10*x - 20)).expand() + 0 + sage: 2*w - ((3*pi^2 - 2*pi - 10)^2/(4*pi^3 - 4*pi^2 - 40*pi - 79) - 2*pi + 1 : (sqrt(4*pi^3 - 4*pi^2 - 40*pi - 79) + 1)/2 - (3*pi^2 - 2*pi - 10)*(-(3*pi^2 - 2*pi - 10)^2/(4*pi^3 - 4*pi^2 - 40*pi - 79) + 3*pi - 1)/sqrt(4*pi^3 - 4*pi^2 - 40*pi - 79) - 1 : 1) + (-2*pi + (2*pi - 3*pi^2 + 10)^2/(-40*pi + 4*pi^3 - 4*pi^2 - 79) + 1 : (2*pi - 3*pi^2 + 10)*(3*pi - (2*pi - 3*pi^2 + 10)^2/(-40*pi + 4*pi^3 - 4*pi^2 - 79) - 1)/sqrt(-40*pi + 4*pi^3 - 4*pi^2 - 79) + 1/2*sqrt(-40*pi + 4*pi^3 - 4*pi^2 - 79) - 1/2 : 1) + + sage: x, y, z = 2*w; temp = ((y^2 + y) - (x^3 - x^2 - 10*x - 20)).expand() + + This is not simplified to zero correctly: + + sage: temp + 0 + + But in fact it is zero, as this suggests: + + sage: RDF(temp).abs() < 1e-10 + True """ a = [SR(x) for x in self.a_invariants()] x, y = SR.var('x, y') diff -r 2d4bc47a082f -r d4f2118fb20e sage/symbolic/expression.pyx --- a/sage/symbolic/expression.pyx Sat May 09 11:09:27 2009 -0700 +++ b/sage/symbolic/expression.pyx Sat May 09 13:59:00 2009 -0700 @@ -3154,7 +3154,7 @@ return R({tuple([0]*R.ngens()):self}) else: return R([self]) - return self.polynomial(ring=R) + return self.polynomial(None, ring=R) def power_series(self, base_ring): """ diff -r 2d4bc47a082f -r d4f2118fb20e sage/symbolic/expression_conversions.py --- a/sage/symbolic/expression_conversions.py Sat May 09 11:09:27 2009 -0700 +++ b/sage/symbolic/expression_conversions.py Sat May 09 13:59:00 2009 -0700 @@ -207,10 +207,10 @@ """ raise NotImplementedError - def relation(self, ex): + def relation(self, ex, operator): """ The input to this method is a symbolic expression which - corresponds to a relation. + corresponds to a relation with specified operator. TESTS:: @@ -220,6 +220,10 @@ Traceback (most recent call last): ... NotImplementedError + sage: Converter().relation(x==3, operator.lt) + Traceback (most recent call last): + ... + NotImplementedError """ raise NotImplementedError @@ -318,6 +322,8 @@ sage: m = InterfaceInit(maxima) sage: m.relation(x==3, operator.eq) 'x = 3' + sage: m.relation(x==3, operator.lt) + 'x < 3' """ return "%s %s %s"%(self(ex.lhs()), self.relation_symbols[operator], self(ex.rhs())) @@ -642,6 +648,33 @@ """ return self.base_ring(ex) + def relation(self, ex, op): + """ + EXAMPLES:: + + sage: import operator + sage: from sage.symbolic.expression_conversions import PolynomialConverter + + sage: x, y = var('x, y') + sage: p = PolynomialConverter(x, base_ring=RR) + + sage: p.relation(x==3, operator.eq) + 1.00000000000000*x - 3.00000000000000 + sage: p.relation(x==3, operator.lt) + Traceback (most recent call last): + ... + ValueError: Unable to represent as a polynomial + + sage: p = PolynomialConverter(x - y, base_ring=QQ) + sage: p.relation(x^2 - y^3 + 1 == x^3, operator.eq) + -x^3 - y^3 + x^2 + 1 + """ + import operator + if op == operator.eq: + return self(ex.lhs()) - self(ex.rhs()) + else: + raise ValueError, "Unable to represent as a polynomial" + def arithmetic(self, ex, operator): if len(ex.variables()) == 0: return self.base_ring(ex) @@ -674,6 +707,12 @@ sage: _.parent() Multivariate Polynomial Ring in x, y over Rational Field + sage: x, y = var('x, y') + sage: polynomial(x + y^2, ring=QQ['x,y']) + y^2 + x + sage: _.parent() + Multivariate Polynomial Ring in x, y over Rational Field + The polynomials can have arbitrary (constant) coefficients so long as they coerce into the base ring::