Timeline for Polynomial that is not always a square over $\mathbb{Z}_p$
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Jun 1, 2023 at 15:27 | comment | added | Peter Mueller | Yes, if $d$ is the degree of an absolutely irreducible curve with $N$ points in $\mathbb F_p^2$, then $\lvert N-p-1\rvert\le (d-1)(d-2)\sqrt{p}+d$, see e.g. Theorem 5.4.1 in the book "Field Arithmetic" by Fried and Jarden. | |
| Jun 1, 2023 at 15:02 | comment | added | Dom | Thank you! More generally, can we use a similar argument if we replace the cubic power by a larger power, say (1+x^2)^t-1? Would we have a growing number of cases to verify manually? The cases t = (p-1)/2 and t = p-1 are special and can be resolved directly. When $t \ne (p-1)/2, p-1$, I always find a lot of solutions for any p. | |
| Jun 1, 2023 at 14:48 | vote | accept | Dom | ||
| Jun 1, 2023 at 14:27 | history | answered | Noam D. Elkies | CC BY-SA 4.0 |