On the Győry-Sárközy-Stewart conjecture in function fields

Igor E. Shparlinski

Czechoslovak Mathematical Journal (2018)

  • Volume: 68, Issue: 4, page 1067-1077
  • ISSN: 0011-4642

Abstract

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We consider function field analogues of the conjecture of Győry, Sárközy and Stewart (1996) on the greatest prime divisor of the product ( a b + 1 ) ( a c + 1 ) ( b c + 1 ) for distinct positive integers a , b and c . In particular, we show that, under some natural conditions on rational functions F , G , H ∈ ℂ ( X ) , the number of distinct zeros and poles of the shifted products F H + 1 and G H + 1 grows linearly with deg H if deg H ≥ max { deg F , deg G } . We also obtain a version of this result for rational functions over a finite field.

How to cite

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Shparlinski, Igor E.. "On the Győry-Sárközy-Stewart conjecture in function fields." Czechoslovak Mathematical Journal 68.4 (2018): 1067-1077. <http://eudml.org/doc/294717>.

@article{Shparlinski2018,
abstract = {We consider function field analogues of the conjecture of Győry, Sárközy and Stewart (1996) on the greatest prime divisor of the product $(ab+1)(ac+1)(bc+1)$ for distinct positive integers $a$, $b$ and $c$. In particular, we show that, under some natural conditions on rational functions $F,G,H \in \{\mathbb \{C\}\}(X)$, the number of distinct zeros and poles of the shifted products $FH+1$ and $GH+1$ grows linearly with $\deg H$ if $\deg H \ge \max \lbrace \deg F, \deg G\rbrace $. We also obtain a version of this result for rational functions over a finite field.},
author = {Shparlinski, Igor E.},
journal = {Czechoslovak Mathematical Journal},
keywords = {shifted polynomial product; number of zeros},
language = {eng},
number = {4},
pages = {1067-1077},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the Győry-Sárközy-Stewart conjecture in function fields},
url = {http://eudml.org/doc/294717},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Shparlinski, Igor E.
TI - On the Győry-Sárközy-Stewart conjecture in function fields
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 4
SP - 1067
EP - 1077
AB - We consider function field analogues of the conjecture of Győry, Sárközy and Stewart (1996) on the greatest prime divisor of the product $(ab+1)(ac+1)(bc+1)$ for distinct positive integers $a$, $b$ and $c$. In particular, we show that, under some natural conditions on rational functions $F,G,H \in {\mathbb {C}}(X)$, the number of distinct zeros and poles of the shifted products $FH+1$ and $GH+1$ grows linearly with $\deg H$ if $\deg H \ge \max \lbrace \deg F, \deg G\rbrace $. We also obtain a version of this result for rational functions over a finite field.
LA - eng
KW - shifted polynomial product; number of zeros
UR - http://eudml.org/doc/294717
ER -

References

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