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I am looking for the standard term for a system that consists of things of the form $p_i(x_1,\ldots ,x_n)=0$ and of the form $q_j(x_1,\ldots,x_n)\neq 0$ with the $p_i$ and $q_j$ polynomials. I have seen used the term "system of equations and inequalities" but to me inequality means "$\leq$" or "$\geq$" and not "$\neq$" so I would prefer not to use it. Is there some commonly accepted term?

I made this CW since I am not sure there is a correct answer.

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See "semialgebraic set" ... ... in that setting, of course, $q_j \ne 0$ is the same as $q_j^2 > 0$. – Gerald Edgar Apr 7 '12 at 17:55
Also search for inequations. Gerhard "Ask Me About System Design" Paseman, 2012.04.07 – Gerhard Paseman Apr 7 '12 at 18:19
I've also seen the term "disequality" or "disequation". – Andreas Blass Apr 7 '12 at 18:23
These are called "equations and inequations" in the literature related to first order theories (say, Tarski problem and such). – Mark Sapir Apr 7 '12 at 21:21
There's also the standard trick of introducing auxiliary variables $y_j$ and replacing each nequation $q_j(x_1,\ldots,x_n) \neq 0$ by the equation $q_j(x_1,\ldots,x_n) \cdot y_j = 1$. – Noam D. Elkies Apr 8 '12 at 0:53

In algebraic geometry, it's called a quasi-projective algebraic set, which by definition is a Zariski open subset of a Zariski closed subset of projective space. (I'm assuming you only have finitely many $p_i$'s and finitely many $q_j$'s.) Since you're using non-homogeneous polynomials, you're starting in affine space, but that's simply projective space in the homogeneous variables $X_0,\ldots,X_n$ with the condition $X_0\ne0$, so it fits into your framework.

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Thanks Joe. I'm really looking though for the name of the set of equations and "anti"-equations rather than the set of points in affine space which they define. – Benjamin Steinberg Apr 7 '12 at 20:48

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