New bounds for the number of connected components of fewnomial hypersurfaces
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866910436400562176 |
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| author | Bihan, Frédéric Humbert, Tristan Tavenas, Sébastien |
| author_facet | Bihan, Frédéric Humbert, Tristan Tavenas, Sébastien |
| contents | We prove that the number of connected components of a smooth hypersurface in the positive orthant of $\mathbb{R}^n$ defined by a real polynomial with $d + k + 1$ monomials, where $d$ is the dimension of the affine span of the exponent vectors, is smaller than or equal to $8(d+1)^{k-1} 2^{k-1 \choose 2}$, improving the previously known bounds. We refine this bound for $k = 2$ by showing that a smooth hypersurface defined by a real polynomial with $d+3$ monomials in $n$ variables has at most $\lfloor(d-1)/2\rfloor + 3$ connected components in the positive orthant of $\mathbb{R}^n$. We present an explicit polynomial in $2$ variables with $5$ monomials which defines a curve with three connected components in the positive orthant, showing that our bound is sharp for $d = 2$ (and any $n$). Our results hold for polynomials with real exponent vectors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_04590 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | New bounds for the number of connected components of fewnomial hypersurfaces Bihan, Frédéric Humbert, Tristan Tavenas, Sébastien Algebraic Geometry We prove that the number of connected components of a smooth hypersurface in the positive orthant of $\mathbb{R}^n$ defined by a real polynomial with $d + k + 1$ monomials, where $d$ is the dimension of the affine span of the exponent vectors, is smaller than or equal to $8(d+1)^{k-1} 2^{k-1 \choose 2}$, improving the previously known bounds. We refine this bound for $k = 2$ by showing that a smooth hypersurface defined by a real polynomial with $d+3$ monomials in $n$ variables has at most $\lfloor(d-1)/2\rfloor + 3$ connected components in the positive orthant of $\mathbb{R}^n$. We present an explicit polynomial in $2$ variables with $5$ monomials which defines a curve with three connected components in the positive orthant, showing that our bound is sharp for $d = 2$ (and any $n$). Our results hold for polynomials with real exponent vectors. |
| title | New bounds for the number of connected components of fewnomial hypersurfaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2208.04590 |