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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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    $\begingroup$ See "semialgebraic set" ... en.wikipedia.org/wiki/Semialgebraic_set ... in that setting, of course, $q_j \ne 0$ is the same as $q_j^2 > 0$. $\endgroup$
    – Gerald Edgar
    Commented Apr 7, 2012 at 17:55
  • $\begingroup$ Also search for inequations. Gerhard "Ask Me About System Design" Paseman, 2012.04.07 $\endgroup$
    – Gerhard Paseman
    Commented Apr 7, 2012 at 18:19
  • $\begingroup$ I've also seen the term "disequality" or "disequation". $\endgroup$
    – Andreas Blass
    Commented Apr 7, 2012 at 18:23
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    $\begingroup$ These are called "equations and inequations" in the literature related to first order theories (say, Tarski problem and such). $\endgroup$
    – user6976
    Commented Apr 7, 2012 at 21:21
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    $\begingroup$ 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 $q_j(x_1,\ldots,x_n) \cdot y_j = 1$. $\endgroup$
    – Noam D. Elkies
    Commented Apr 8, 2012 at 0:53

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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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  • $\begingroup$ 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. $\endgroup$
    – Benjamin Steinberg
    Commented Apr 7, 2012 at 20:48

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