Geodesic
On the Győry-Sárközy-Stewart conjecture in function fields
Czechoslovak Mathematical Journal, Tome 68 (2018) no. 4, pp. 1067-1077 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $(ab+1)(ac+1)(bc+1)$ 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 \{\deg F, \deg G\} $. We also obtain a version of this result for rational functions over a finite field.
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 \{\deg F, \deg G\} $. We also obtain a version of this result for rational functions over a finite field.
DOI : 10.21136/CMJ.2018.0085-17
Classification : 11R09, 11S05, 12E05
Keywords: shifted polynomial product; number of zeros 
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     title = {On the {Gy\H{o}ry-S\'ark\"ozy-Stewart} conjecture in function fields},
     journal = {Czechoslovak Mathematical Journal},
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     year = {2018},
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Shparlinski, Igor E. On the Győry-Sárközy-Stewart conjecture in function fields. Czechoslovak Mathematical Journal, Tome 68 (2018) no. 4, pp. 1067-1077. doi: 10.21136/CMJ.2018.0085-17

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