Sparsity of the intersection of polynomial images of an interval

Mei-Chu Chang

Mei-Chu Chang

Acta Arithmetica, Tome 166 (2014), p. 243-249 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $f (x), g (x) \in_{p} [x]$ 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 $| f ([0, M]) \cap g ([0, M]) | ≲ M^{1 - ε}$ .

Publié le : 2014-01-01

Zbl 1308.11085

EUDML-ID : urn:eudml:doc:279665

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     author = {Mei-Chu Chang},
     title = {Sparsity of the intersection of polynomial images of an interval},
     journal = {Acta Arithmetica},
     volume = {166},
     year = {2014},
     pages = {243-249},
     zbl = {1308.11085},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-3-3}
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Mei-Chu Chang. Sparsity of the intersection of polynomial images of an interval. Acta Arithmetica, Tome 166 (2014) pp. 243-249. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-3-3/