判别式、对称化图单项式与平方和
Discriminants, symmetrized graph monomials, and sums of squares
摘要：
受J.J.Sylvester在1878年提出的二元形式不变理论的启发，对于每个可能有重边但没有环的图，给出了它的对称化图单项式，它是原图的顶点标号中的多项式。在20世纪，这个结构被几位作者研究过。我们提出的问题，哪些图这个多项式是一个非负的响应。平方和。这个问题是由F. Sottile和E. Mukhin关于一元多项式导数的判别式，以及P.和A.对称化图单项非负但不是平方和的4边图的Lax。我们提出了详细的信息对称图单项式的图与四个和六个边缘，通过计算机计算。
英文摘要：
Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed in 1878 for each graph with possible multiple edges but without loops its symmetrized graph monomial which is a polynomial in the vertex labels of the original graph. In the 20-th century this construction was studied by several authors. We pose the question for which graphs this polynomial is a non-negative resp. a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on discriminant of the derivative of a univariate polynomial, and an interesting example of P. and A. Lax of a graph with 4 edges whose symmetrized graph monomial is non-negative but not a sum of squares. We present detailed information about symmetrized graph monomials for graphs with four and six edges, obtained by computer calculations.
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