degenerate polynomial
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10.37236/8668 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Mehdi Makhul ◽  
Oliver Roche-Newton ◽  
Audie Warren ◽  
Frank De Zeeuw

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.


2017 ◽  
Vol 42 (3) ◽  
pp. 563-585
Author(s):  
Si Tiep Dinh ◽  
Huy Vui Ha ◽  
Tien Son Pham

2014 ◽  
Vol 410 (2) ◽  
pp. 541-560 ◽  
Author(s):  
Sĩ Tiệp Đinh ◽  
Huy Vui Hà ◽  
Tiến Sơn Phạm ◽  
Nguyễn Thị Tha̓o

2003 ◽  
Vol 172 ◽  
pp. 31-58 ◽  
Author(s):  
W. A. Zuniga-Galindo

AbstractTo a polynomial f over a non-archimedean local field K and a character χ of the group of units of the valuation ring of K one associates Igusa’s local zeta function Z (s, f, χ). In this paper, we study the local zeta function Z(s, f, χ) associated to a non-degenerate polynomial f, by using an approach based on the p-adic stationary phase formula and Néron p-desingularization. We give a small set of candidates for the poles of Z (s, f, χ) in terms of the Newton polyhedron Γ(f) of f. We also show that for almost all χ, the local zeta function Z(s, f, χ) is a polynomial in q−s whose degree is bounded by a constant independent of χ. Our second result is a description of the largest pole of Z(s, f, χtriv) in terms of Γ(f) when the distance between Γ(f) and the origin is at most one.


1993 ◽  
Vol 5 (11) ◽  
pp. 557-568 ◽  
Author(s):  
Bas J. Hogervorst ◽  
Ruud van Damme

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