Sparsity of the intersection of polynomial images of an interval

Mei-Chu Chang. "Sparsity of the intersection of polynomial images of an interval." Acta Arithmetica 165.3 (2014): 243-249. <http://eudml.org/doc/279665>.

@article{Mei2014,
abstract = {We show that the intersection of the images of two polynomial maps on a given interval is sparse. More precisely, we prove the following. Let $f(x),g(x) ∈ _\{p\}[x]$ be polynomials of degrees d and e with d ≥ e ≥ 2. Suppose M ∈ ℤ satisfies $p^\{1/E(1 + κ/(1-κ)\} > M > p^ε$, where E = e(e+1)/2 and κ = (1/d - 1/d²) (E-1)/E + ε. Assume f(x)-g(y) is absolutely irreducible. Then $|f([0,M]) ∩ g([0, M])| ≲ M^\{1-ε\}$.},
author = {Mei-Chu Chang},
journal = {Acta Arithmetica},
keywords = {counting solutions; congruence equations; lattice points; character sums},
language = {eng},
number = {3},
pages = {243-249},
title = {Sparsity of the intersection of polynomial images of an interval},
url = {http://eudml.org/doc/279665},
volume = {165},
year = {2014},
}

TY - JOUR
AU - Mei-Chu Chang
TI - Sparsity of the intersection of polynomial images of an interval
JO - Acta Arithmetica
PY - 2014
VL - 165
IS - 3
SP - 243
EP - 249
AB - We show that the intersection of the images of two polynomial maps on a given interval is sparse. More precisely, we prove the following. Let $f(x),g(x) ∈ _{p}[x]$ be polynomials of degrees d and e with d ≥ e ≥ 2. Suppose M ∈ ℤ satisfies $p^{1/E(1 + κ/(1-κ)} > M > p^ε$, where E = e(e+1)/2 and κ = (1/d - 1/d²) (E-1)/E + ε. Assume f(x)-g(y) is absolutely irreducible. Then $|f([0,M]) ∩ g([0, M])| ≲ M^{1-ε}$.
LA - eng
KW - counting solutions; congruence equations; lattice points; character sums
UR - http://eudml.org/doc/279665
ER -