Computing a determinantal representation of a bivariate polynomial - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T12:24:19Z http://mathoverflow.net/feeds/question/91627 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91627/computing-a-determinantal-representation-of-a-bivariate-polynomial Computing a determinantal representation of a bivariate polynomial alext87 2012-03-19T15:46:52Z 2012-10-24T08:52:41Z <p>Let \$p(x,y,z)\$ be a homogeneous irreducible polynomial of degree \$d\$, with real coefficients. From Dickson in 1920 we know that there exists \$A\$, \$B\$ and \$C\$ such that </p> <p>\$ det(Ax + By + Cz) = c p(x,y,z)\$,</p> <p>where \$c\$ is some constant. </p> <p>Vinnokov in 1988 was able to describe all the non-equivalent determinantal representations as points on the Jacobian variety that are not on the exceptional sub variety. The theoretical work in this paper is relatively constructive, but is still a long way from a numerically stable constructive algorithm for \$A\$, \$B\$ and \$C\$. </p> <p>Given any polynomial \$p(x,y,z)\$, can one triple \$(A, B,C)\$ be constructed in a numerically stable way?</p> <p>Thanks in advance. </p> http://mathoverflow.net/questions/91627/computing-a-determinantal-representation-of-a-bivariate-polynomial/110524#110524 Answer by Markus Schweighofer for Computing a determinantal representation of a bivariate polynomial Markus Schweighofer 2012-10-24T08:52:41Z 2012-10-24T08:52:41Z <p>This is discussed in a recent work of Plaumann, Sturmfels and Vinzant:</p> <p><a href="http://arxiv.org/abs/1011.6057" rel="nofollow">http://arxiv.org/abs/1011.6057</a></p>