Timeline for Nonvanishing criterion for a polynomial of polynomials
Current License: CC BY-SA 4.0
6 events
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| Sep 11, 2020 at 23:49 | comment | added | Thurmond | @QiaochuYuan Sorry, I posted in the middle of editing. I need to think a little bit about enlarging the interpretation (I1); there is a rough analogy with the divisibility predicates necessary to do quantifier elimination in Presburger arithmetic that I am looking to preserve. Your comment was very helpful in getting me to think about this! | |
| Sep 11, 2020 at 23:46 | comment | added | Qiaochu Yuan | Isn't that a counterexample then? | |
| Sep 11, 2020 at 23:44 | comment | added | Thurmond | @QiaochuYuan The interpretation I had in mind (which I'll denote (I1)) allows only vanishing or non-vanishing criteria on polynomials in the $Q_i$, so your example would not count. This interpretation makes the question seem like more of a pure commutative-algebra question. There is a more permissive alternative (I2), which is to consider any first-order sentence in $\mathrm{Th}(\mathbb{F}_q[X], +, \times, 0, 1, X)$ in the variables $Q_i$. I think this is too permissive for my tastes, but perhaps a slight enlargement of (I1) would be allowable. I'll have to think. | |
| Sep 7, 2020 at 0:21 | comment | added | Qiaochu Yuan | What is an "algebraic criterion" to you? For example, consider $n = 1, P(x, y) = y^2 - x$, so that $P(Q(x), y) = y^2 - Q(x)$. Then your condition is that $Q(x)$ doesn't admit a square root; does this count? | |
| Sep 6, 2020 at 14:11 | review | First posts | |||
| Sep 6, 2020 at 17:05 | |||||
| Sep 6, 2020 at 14:06 | history | asked | Thurmond | CC BY-SA 4.0 |