Number of integers represented by families of binary forms III: fewnomials
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| Format: | Preprint |
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2025
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| _version_ | 1866916944572055552 |
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| author | Fouvry, Etienne Waldschmidt, Michel |
| author_facet | Fouvry, Etienne Waldschmidt, Michel |
| contents | In a series of papers we investigated the following question: given a family $\calF$ of binary forms having nonzero discriminant and integer coefficients, for each $d\geqslant 3$, we estimate the number of integers $m$ with $|m|\leqslant N$ which are represented by an element in $\calF$ of degree $\geqslant d$. Under suitable assumptions, asymptotically as $N\to\infty$, the main term in the estimate is given by the forms in $\calF$ having degree $d$ (if any), while the forms of degree $>d$ contribute only to the error term. The present text is devoted to fewnomials \[ a_0X^{kr}+a_1X^{k(r-1)}Y^k+\cdots +a_{r-1}X^kY^{k(r-1)}+a_rY^{kr} \] with fixed $r\geqslant 1$ and varying $k,a_0,a_1,\dots,a_r$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_08335 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Number of integers represented by families of binary forms III: fewnomials Fouvry, Etienne Waldschmidt, Michel Number Theory Primary 11E76, Secondary 11D45 11D85 In a series of papers we investigated the following question: given a family $\calF$ of binary forms having nonzero discriminant and integer coefficients, for each $d\geqslant 3$, we estimate the number of integers $m$ with $|m|\leqslant N$ which are represented by an element in $\calF$ of degree $\geqslant d$. Under suitable assumptions, asymptotically as $N\to\infty$, the main term in the estimate is given by the forms in $\calF$ having degree $d$ (if any), while the forms of degree $>d$ contribute only to the error term. The present text is devoted to fewnomials \[ a_0X^{kr}+a_1X^{k(r-1)}Y^k+\cdots +a_{r-1}X^kY^{k(r-1)}+a_rY^{kr} \] with fixed $r\geqslant 1$ and varying $k,a_0,a_1,\dots,a_r$. |
| title | Number of integers represented by families of binary forms III: fewnomials |
| topic | Number Theory Primary 11E76, Secondary 11D45 11D85 |
| url | https://arxiv.org/abs/2509.08335 |