On fewnomials, integral points and a toric version of Bertini's theorem
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| Format: | Preprint |
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2014
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| _version_ | 1866914648061640704 |
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| author | Fuchs, Clemens Mantova, Vincenzo Zannier, Umberto |
| author_facet | Fuchs, Clemens Mantova, Vincenzo Zannier, Umberto |
| contents | An old conjecture of Erdős and Rényi, proved by Schinzel, predicted a bound for the number of terms of a polynomial $g(x) \in \mathbb{C}[x]$ when its square $g(x)^2$ has a given number of terms. Further conjectures and results arose, but some fundamental questions remained open.
In this paper, with methods which appear to be new, we achieve a final result in this direction for completely general algebraic equations $f(x,g(x))=0$, where $f(x,y)$ is monic of arbitrary degree in $y$, and has boundedly many terms in $x$: we prove that the number of terms of such a $g(x)$ is necessarily bounded. This includes the previous results as extremely special cases.
We shall interpret polynomials with boundedly many terms as the restrictions to 1-parameter subgroups or cosets of regular functions of bounded degree on a given torus $\mathbb{G}_{\mathrm{m}}^l$. Such a viewpoint shall lead to some best-possible corollaries in the context of finite covers of $\mathbb{G}_{\mathrm{m}}^l$, concerning the structure of their integral points over function fields (in the spirit of conjectures of Vojta) and a Bertini-type irreducibility theorem above algebraic multiplicative cosets. A further natural reading occurs in non-standard arithmetic, where our result translates into an algebraic and integral-closedness statement inside the ring of non-standard polynomials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1412_4548 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | On fewnomials, integral points and a toric version of Bertini's theorem Fuchs, Clemens Mantova, Vincenzo Zannier, Umberto Number Theory 11C08 12E05 12Y05 14G05 14J99 11U10 An old conjecture of Erdős and Rényi, proved by Schinzel, predicted a bound for the number of terms of a polynomial $g(x) \in \mathbb{C}[x]$ when its square $g(x)^2$ has a given number of terms. Further conjectures and results arose, but some fundamental questions remained open. In this paper, with methods which appear to be new, we achieve a final result in this direction for completely general algebraic equations $f(x,g(x))=0$, where $f(x,y)$ is monic of arbitrary degree in $y$, and has boundedly many terms in $x$: we prove that the number of terms of such a $g(x)$ is necessarily bounded. This includes the previous results as extremely special cases. We shall interpret polynomials with boundedly many terms as the restrictions to 1-parameter subgroups or cosets of regular functions of bounded degree on a given torus $\mathbb{G}_{\mathrm{m}}^l$. Such a viewpoint shall lead to some best-possible corollaries in the context of finite covers of $\mathbb{G}_{\mathrm{m}}^l$, concerning the structure of their integral points over function fields (in the spirit of conjectures of Vojta) and a Bertini-type irreducibility theorem above algebraic multiplicative cosets. A further natural reading occurs in non-standard arithmetic, where our result translates into an algebraic and integral-closedness statement inside the ring of non-standard polynomials. |
| title | On fewnomials, integral points and a toric version of Bertini's theorem |
| topic | Number Theory 11C08 12E05 12Y05 14G05 14J99 11U10 |
| url | https://arxiv.org/abs/1412.4548 |