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On the smallest integer vector at which a multivariable polynomial does not vanish

Arturas Dubickas   ORCID icon orcid.org/0000-0002-3625-9466 ; Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania

Puni tekst: engleski, pdf (106 KB) str. 227-235 preuzimanja: 52* citiraj
APA 6th Edition
Dubickas, A. (2020). On the smallest integer vector at which a multivariable polynomial does not vanish. Mathematical Communications, 25 (2), 227-235. Preuzeto s https://hrcak.srce.hr/244263
MLA 8th Edition
Dubickas, Arturas. "On the smallest integer vector at which a multivariable polynomial does not vanish." Mathematical Communications, vol. 25, br. 2, 2020, str. 227-235. https://hrcak.srce.hr/244263. Citirano 21.06.2021.
Chicago 17th Edition
Dubickas, Arturas. "On the smallest integer vector at which a multivariable polynomial does not vanish." Mathematical Communications 25, br. 2 (2020): 227-235. https://hrcak.srce.hr/244263
Harvard
Dubickas, A. (2020). 'On the smallest integer vector at which a multivariable polynomial does not vanish', Mathematical Communications, 25(2), str. 227-235. Preuzeto s: https://hrcak.srce.hr/244263 (Datum pristupa: 21.06.2021.)
Vancouver
Dubickas A. On the smallest integer vector at which a multivariable polynomial does not vanish. Mathematical Communications [Internet]. 2020 [pristupljeno 21.06.2021.];25(2):227-235. Dostupno na: https://hrcak.srce.hr/244263
IEEE
A. Dubickas, "On the smallest integer vector at which a multivariable polynomial does not vanish", Mathematical Communications, vol.25, br. 2, str. 227-235, 2020. [Online]. Dostupno na: https://hrcak.srce.hr/244263. [Citirano: 21.06.2021.]

Sažetak
We prove that for any polynomial $P$ of degree $d$ in $\C[x_1,\dots,x_n]$ there exists a vector $(u_1,\dots,u_n) \in \Z^n$ such that $P(u_1,\dots,u_n) \ne 0$ and $\sum_{i=1}^n |u_i| \leq \min\{d, \lfloor (d+n)/2 \rfloor\}$. We also show that this bound is best possible. Similarly, for any $P \in \C[x_1,\dots,x_n]$ of degree $d$ and any real number $p \geq \log 3/\log 2$ there is a vector $(u_1,\dots,u_n) \in \Z^n$ such that $P(u_1,\dots,u_n) \ne 0$ and $\sum_{i=1}^n |u_i|^p \leq \max\{1+\lfloor d/2 \rfloor^p, \lfloor (d+1)/2 \rfloor^p\}$. The latter bound is also best possible for every $n \geq 2$.

Ključne riječi
multivariable polynomial; combinatorial Nullstellensatz; $L^p$-norm; integer vector

Hrčak ID: 244263

URI
https://hrcak.srce.hr/244263

Posjeta: 132 *