Improved fewnomial upper bounds from Wronskians and dessins d'enfant
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913490534400000 |
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| author | Hilany, Boulos El Tavenas, Sébastien |
| author_facet | Hilany, Boulos El Tavenas, Sébastien |
| contents | We use Grothendieck's dessins d'enfant to show that if $P$ and $Q$ are two real polynomials, any real function of the form $x^α(1-x)^β P - Q$, has at most $°P +°Q + 2$ roots in the interval $]0,~1[$. As a consequence, we obtain an upper bound on the number of positive solutions to a real polynomial system $f=g=0$ in two variables where $f$ has three monomials terms, and $g$ has $t$ terms. The approach we adopt for tackling this Fewnomial bound relies on the theory of Wronskians, which was used in Koiran et.\ al.\ (J.\ Symb.\ Comput., 2015) for producing the first upper bound which is polynomial in $t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01651 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved fewnomial upper bounds from Wronskians and dessins d'enfant Hilany, Boulos El Tavenas, Sébastien Algebraic Geometry Combinatorics Functional Analysis 14P25, 30D30, 14Q99 We use Grothendieck's dessins d'enfant to show that if $P$ and $Q$ are two real polynomials, any real function of the form $x^α(1-x)^β P - Q$, has at most $°P +°Q + 2$ roots in the interval $]0,~1[$. As a consequence, we obtain an upper bound on the number of positive solutions to a real polynomial system $f=g=0$ in two variables where $f$ has three monomials terms, and $g$ has $t$ terms. The approach we adopt for tackling this Fewnomial bound relies on the theory of Wronskians, which was used in Koiran et.\ al.\ (J.\ Symb.\ Comput., 2015) for producing the first upper bound which is polynomial in $t$. |
| title | Improved fewnomial upper bounds from Wronskians and dessins d'enfant |
| topic | Algebraic Geometry Combinatorics Functional Analysis 14P25, 30D30, 14Q99 |
| url | https://arxiv.org/abs/2409.01651 |