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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.09008 |
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| _version_ | 1866908531775504384 |
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| author | Riggs, Brittany |
| author_facet | Riggs, Brittany |
| contents | Let $f$ be a monic univariate polynomial. We say that $f$ is positive if $f(x)$ is positive over all $x > 0$. If all the coefficients of $f$ are non-negative, then $f$ is trivially positive. In 1883, Poincaré proved that $f$ is positive if and only if there exists a monic polynomial $g$ such that all the coefficients of $gf$ are non-negative. Such polynomial $g$ is called a Poincaré multiplier for the positive polynomial $f$. Of course one hopes to find a multiplier with smallest degree. In 1911, Meissner provided such a bound for quadratic polynomials. In this paper, we provide a linear algebra proof of Meissner's optimal bound and compare an improved optimal degree Poincaré multiplier to one provided by Meissner. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09008 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A New Proof of Meissner's Optimal Bound on the Degree of a Poincaré Multiplier and an Improved Optimal Degree Multiplier Riggs, Brittany Algebraic Geometry 12D10 Let $f$ be a monic univariate polynomial. We say that $f$ is positive if $f(x)$ is positive over all $x > 0$. If all the coefficients of $f$ are non-negative, then $f$ is trivially positive. In 1883, Poincaré proved that $f$ is positive if and only if there exists a monic polynomial $g$ such that all the coefficients of $gf$ are non-negative. Such polynomial $g$ is called a Poincaré multiplier for the positive polynomial $f$. Of course one hopes to find a multiplier with smallest degree. In 1911, Meissner provided such a bound for quadratic polynomials. In this paper, we provide a linear algebra proof of Meissner's optimal bound and compare an improved optimal degree Poincaré multiplier to one provided by Meissner. |
| title | A New Proof of Meissner's Optimal Bound on the Degree of a Poincaré Multiplier and an Improved Optimal Degree Multiplier |
| topic | Algebraic Geometry 12D10 |
| url | https://arxiv.org/abs/2509.09008 |