Expansion of trivariate polynomials using proximity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908591708962816 |
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| author | Raz, Orit E. |
| author_facet | Raz, Orit E. |
| contents | We extend the proximity technique of Solymosi and Zahl [J. Combin. Theory, Ser. A (2024)] to the setting of trivariate polynomials. In particular, we prove the following result: Let $f(x,y,z)=(x-y)^2+(φ(x)-z)^2$, where $φ(x)\in \mathbb{R}[x]$ has degree at least 3. Then, for every finite $A,B,C\subset \mathbb{R}$ each of size $n$, one has $|f(A,B,C)|=Ω(n^{5/3-\varepsilon})$, for every $\varepsilon>0$, where the constant of proportionality depends on $\varepsilon$ and on ${\rm deg}(φ)$. This improves the previous exponent $3/2$, due to Raz, Sharir, and De Zeeuw [Israel J. Math. (2018)]. To the best of our knowledge, prior to this work no trivariate polynomial was known to have expansion exceeding $Ω(n^{3/2})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_12191 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Expansion of trivariate polynomials using proximity Raz, Orit E. Combinatorics We extend the proximity technique of Solymosi and Zahl [J. Combin. Theory, Ser. A (2024)] to the setting of trivariate polynomials. In particular, we prove the following result: Let $f(x,y,z)=(x-y)^2+(φ(x)-z)^2$, where $φ(x)\in \mathbb{R}[x]$ has degree at least 3. Then, for every finite $A,B,C\subset \mathbb{R}$ each of size $n$, one has $|f(A,B,C)|=Ω(n^{5/3-\varepsilon})$, for every $\varepsilon>0$, where the constant of proportionality depends on $\varepsilon$ and on ${\rm deg}(φ)$. This improves the previous exponent $3/2$, due to Raz, Sharir, and De Zeeuw [Israel J. Math. (2018)]. To the best of our knowledge, prior to this work no trivariate polynomial was known to have expansion exceeding $Ω(n^{3/2})$. |
| title | Expansion of trivariate polynomials using proximity |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.12191 |