Timeline for Three theorems on the number of nonzero coefficients of a polynomial
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| Aug 8, 2018 at 13:55 | answer | added | Seva | timeline score: 1 | |
| Aug 5, 2018 at 20:02 | comment | added | Vesselin Dimitrov | @Seva: True, the higher dimensional results on bounding the Betti numbers of a fewnomial's zero locus are less precise, and non-sharp, though in the same spirit. Regarding the variant for non-Archimedean local fields, I have to correct myself in that the $\mathbb{F}_q(T)$ case has actually the sharp bound of $q^{m-1}$ on the number of distinct roots of a polynomial with $m$ non-zero coefficients: see the paper Zeros of sparse polynomials over local fields of characteristic $p$, by Bjorn Poonen. Lenstra's reference there addresses the $\mathbb{Q}_p$ variant, but with a non-sharp bound. | |
| Aug 5, 2018 at 19:51 | comment | added | Seva | @VesselinDimitrov: As I see it, a distinguishing feature of the three theorems that I mentioned is their sharpness; the number of certain roots is at most the number of nonzero coefficients less $1$, which is the precise bound. | |
| Aug 5, 2018 at 19:45 | comment | added | Vesselin Dimitrov | A higher dimensional extension of such results is found in Khovanskii's theory of fewnomials. For a summary, I can refer to this question that I have asked mathoverflow.net/questions/214671/… | |
| Aug 5, 2018 at 19:01 | comment | added | Vesselin Dimitrov | There are also complementary statements about the number of $\mathbb{Q}_p$-roots, or the number of $\mathbb{F}_q(T)$-roots that a polynomial with $m$ non-zero terms may have. In either case: $\ll m$, with an implied coefficient depending on the local field at hand. | |
| Aug 5, 2018 at 18:58 | history | edited | Seva | CC BY-SA 4.0 |
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| Aug 5, 2018 at 18:40 | history | asked | Seva | CC BY-SA 4.0 |