Using hyperelliptic curves to find positive polynomials that are not a sum of three squares in R(x, y)
| dc.creator | Mahé, Valéry | |
| dc.date | 2007-03-24 | |
| dc.date | 2007-09-13 | |
| dc.date.accessioned | 2026-07-07T08:29:14Z | |
| dc.date.available | 2026-07-07T08:29:14Z | |
| dc.description | This article deals with a quantitative aspect of Hilbert's seventeenth problem: producing a collection of real polynomials in two variables of degree 8 in one variable which are positive but are not a sum of three squares of rational fractions. As explained by Huisman and Mahe, a given monic squarefree positive polynomial in two variables x and y of degree in y divisible by 4 is a sum of three squares of rational fractions if and only if the jabobian variety of some hyperelliptic curve (associated to P) has an "antineutral" point. Using this criterium, we follow a method developped by Cassels, Ellison and Pfister to solve our problem : at first we show the Mordell-Weil rank of the jacobian variety J associated to some polynomial is zero (this step is done by doing a 2-descent), and then we check that the jacobian variety J has no antineutral torsion point. | |
| dc.description | 63 pages, a proposition has been added (proposition 2.8) | |
| dc.identifier | https://arxiv.org/abs/math/0703722 | |
| dc.identifier | http://arxiv.org/abs/math/0703722 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/137898 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14H40; 14G05; 14H05; 14P99; 14Q05 | |
| dc.title | Using hyperelliptic curves to find positive polynomials that are not a sum of three squares in R(x, y) | |
| dc.type | text |