Is there a way to find out whether a system of polynomial equations has any real solutions? Solve returns complex solutions, which I don't want. For example, I want to be able to discover that (x^2+0.75) + (y^2+0.5) == 1 has no solutions for real x and y. I thought that GroebnerBasis would do the trick, but that doesn't always indicate that a system is unsolvable. For example, GroebnerBasis[x^2+1] just returns 1+x^2, which is true but not very useful for determining (automatically) whether the system of one equation x^2+1 == 0 has any real solutions. I'd like to be able to solve systems of several polynomial equations in several variables.
What I'm really trying to do is find out whether there exists a solution to a (generally nonlinear) polynomial inequality with several variables, subject to linear constraints on the variables. But that can be translated into the above problem by substituting different variables.