Geodesic
Multivariate Polynomial Values in Difference Sets
Discrete analysis (2021) Cet article a éte moissonné depuis la source Scholastica

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For $\ell\geq 2$ and $h\in \mathbb{Z}[x_1,\dots,x_{\ell}]$ of degree $k\geq 2$, we show that every set $A\subseteq \{1,2,\dots,N\}$ lacking nonzero differences in $h(\mathbb{Z}^{\ell})$ satisfies $|A|\ll_h Ne^{-c(\log N)^μ}$, where $c=c(h)>0$, $μ=[(k-1)^2+1]^{-1}$ if $\ell=2$, and $μ=1/2$ if $\ell\geq 3$, provided $h(\mathbb{Z}^{\ell})$ contains a multiple of every natural number and $h$ satisfies certain nonsingularity conditions. We also explore these conditions in detail, drawing on a variety of tools from algebraic geometry.
Publié le : 
@article{DAS_2021_a16,
     author = {John R. Doyle and Alex Rice},
     title = {Multivariate {Polynomial} {Values} in {Difference} {Sets}},
     journal = {Discrete analysis},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2021_a16/}
}
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AU  - John R. Doyle
AU  - Alex Rice
TI  - Multivariate Polynomial Values in Difference Sets
JO  - Discrete analysis
PY  - 2021
UR  - http://geodesic.mathdoc.fr/item/DAS_2021_a16/
LA  - en
ID  - DAS_2021_a16
ER  - 
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%A John R. Doyle
%A Alex Rice
%T Multivariate Polynomial Values in Difference Sets
%J Discrete analysis
%D 2021
%U http://geodesic.mathdoc.fr/item/DAS_2021_a16/
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%F DAS_2021_a16
John R. Doyle; Alex Rice. Multivariate Polynomial Values in Difference Sets. Discrete analysis (2021). http://geodesic.mathdoc.fr/item/DAS_2021_a16/