تقرير
Factoring multivariate polynomials over hyperfields and the multivariable Descartes' problem
| العنوان: | Factoring multivariate polynomials over hyperfields and the multivariable Descartes' problem |
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| المؤلفون: | Gross, Andreas, Gunn, Trevor |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Algebraic Geometry, Mathematics - Rings and Algebras, 14C17 (primary), 14Q30, 14T99, 16Y20 (secondary) |
| الوصف: | We develop several notions of multiplicity for linear factors of multivariable polynomials over different arithmetics (hyperfields). The key example is multiplicities over the hyperfield of signs, which encapsulates the arithmetic of $\mathbf{R}/\mathbf{R}_{>0}$. These multiplicities give us various upper and lower bounds on the number of linear factors with a given sign pattern in terms of the signs of the coefficients of the factored polynomial. Using resultants, we can transform a square system of polynomials into a single polynomial whose multiplicities give us bounds on the number of positive solutions to the system. In particular, we are able to re-derive the lower bound of Itenberg and Roy on any potential upper bound for the number of solutions to a system of equations with a given sign pattern. In addition, our techniques also explain a particular counterexample of Li and Wang to Itenberg and Roy's proposed upper bound. Comment: 44 pages, 10 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2307.09400 |
| رقم الأكسشن: | edsarx.2307.09400 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |