Discriminants, symmetrized graph monomials, and sums of squares

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

Low temperature magnetic phase diagram of the cubic non-Fermi liquid system CeIn_(3-x)Sn_x

In this paper we report a comprehensive study of the magnetic susceptibility (\chi), resistivity (\rho), and specific heat (C_P), down to 0.5 K of the cubic CeIn_(3-x)Sn_x alloy. The ground state of this system evolves from antiferromagnetic (AF) in CeIn_3(T_N=10.2 K) to intermediate-valent in CeSn_3, and represents the first example of a Ce-lattice cubic non-Fermi liquid (NFL) system where T_N(x) can be traced down to T=0 over more than a decade of temperature. Our results indicate that the disappearance of the AF state occurs near x_c ~ 0.7, although already at x ~ 0.4 significant modifications of the magnetic ground state are observed. Between these concentrations, clear NFL signatures are observed, such as \rho(T)\approx \rho_0 + A T^n (with n<1.5) and C_P(T)\propto -T ln(T) dependencies. Within the ordered phase a first order phase transition occurs for 0.25 < x < 0.5. With larger Sn doping, different weak \rho(T) dependencies are observed at low temperatures between x=1 and x=3 while C_P/T shows only a weak temperature dependence.Comment: 7 pages, 7 figures. Accepted in Eur. J. Phys.