Constructions for the Elekes-Szabó and Elekes-Rónyai problems
The electronic journal of combinatorics, Tome 27 (2020) no. 1
We give a construction of a non-degenerate polynomial $F\in \mathbb R[x,y,z]$ and a set $A$ of cardinality $n$ such that $F$ vanishes on $\Omega(n^{3/2})$ points of $A \times A \times A$, thus providing a new lower bound construction for the Elekes–Szabó problem. We also give a related construction for the Elekes–Rónyai problem restricted to a subgraph. This consists of a polynomial $f\in \mathbb R[x,y]$ that is not additive or multiplicative, a set $A$ of size $n$, and a subset $P\subset A\times A$ of size $\Omega(n^{3/2})$ on which $f$ takes only $n$ distinct values.
@article{10_37236_8668,
author = {Mehdi Makhul and Oliver Roche-Newton and Audie Warren and Frank de Zeeuw},
title = {Constructions for the {Elekes-Szab\'o} and {Elekes-R\'onyai} problems},
journal = {The electronic journal of combinatorics},
year = {2020},
volume = {27},
number = {1},
doi = {10.37236/8668},
zbl = {1439.52017},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8668/}
}
TY - JOUR AU - Mehdi Makhul AU - Oliver Roche-Newton AU - Audie Warren AU - Frank de Zeeuw TI - Constructions for the Elekes-Szabó and Elekes-Rónyai problems JO - The electronic journal of combinatorics PY - 2020 VL - 27 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.37236/8668/ DO - 10.37236/8668 ID - 10_37236_8668 ER -
%0 Journal Article %A Mehdi Makhul %A Oliver Roche-Newton %A Audie Warren %A Frank de Zeeuw %T Constructions for the Elekes-Szabó and Elekes-Rónyai problems %J The electronic journal of combinatorics %D 2020 %V 27 %N 1 %U http://geodesic.mathdoc.fr/articles/10.37236/8668/ %R 10.37236/8668 %F 10_37236_8668
Mehdi Makhul; Oliver Roche-Newton; Audie Warren; Frank de Zeeuw. Constructions for the Elekes-Szabó and Elekes-Rónyai problems. The electronic journal of combinatorics, Tome 27 (2020) no. 1. doi: 10.37236/8668
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