Small clique number graphs with three trivial critical ideals

The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. In this article we provide a set of minimal forbidden graphs for the set of graphs with at most three trivial critical ideals. Then we use these forbidden graphs to characterize the graphs with at most three trivial critical ideals and clique number equal to 2 and 3.Comment: 33 pages, 3 figure

Computing the differential Galois group of a parameterized second-order linear differential equation

We develop algorithms to compute the differential Galois group $G$ associated to a parameterized second-order homogeneous linear differential equation of the form $$\tfrac{\partial^2}{\partial x^2} Y + r_1 \tfrac{\partial}{\partial x} Y + r_0 Y = 0,$$ where the coefficients $r_1, r_0 \in F(x)$ are rational functions in $x$ with coefficients in a partial differential field $F$ of characteristic zero. Our work relies on the procedure developed by Dreyfus to compute $G$ under the assumption that $r_1 = 0$. We show how to complete this procedure to cover the cases where $r_1 \neq 0$, by reinterpreting a classical change of variables procedure in Galois-theoretic terms.Comment: 14 page

Singlet-Doublet Dirac Dark Matter

We analyze a simple extension of the Standard Model where the dark matter particle is a Dirac fermion that is mixture of a singlet and an SU(2) doublet. The model contains only four free parameters: the singlet and the doublet masses and two new Yukawa couplings. Direct detection bounds in this model are very strong and require the dark matter particle to be singlet-like. As a result, its relic density has to be obtained via coannihilations with the doublet. We find that the dark matter mass should be below 750 GeV, that the singlet-doublet mass difference cannot exceed 9%, and that direct detection experiments offer the best chance to probe this scenario. Finally, we also show that this model can effectively arise in well-motivated extensions of the Standard Model.Comment: 14 page

Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation

We propose a new method to compute the unipotent radical $R_u(H)$ of the differential Galois group $H$ associated to a parameterized second-order homogeneous linear differential equation of the form $$\tfrac{\partial^2}{\partial x^2}Y-qY=0,$$ where $q \in F(x)$ is a rational function in $x$ with coefficients in a $\Pi$-field $F$ of characteristic zero, and $\Pi$ is a commuting set of parametric derivations. The procedure developed by Dreyfus reduces the computation of $R_u(H)$ to solving a creative telescoping problem, whose effective solution requires the assumption that the maximal reductive quotient $H / R_u(H)$ is a $\Pi$-constant linear differential algebraic group. When this condition is not satisfied, we compute a new set of parametric derivations $\Pi'$ such that the associated differential Galois group $H'$ has the property that $H'/ R_u(H')$ is $\Pi'$-constant, and such that $R_u(H)$ is defined by the same differential equations as $R_u(H')$. Thus the computation of $R_u(H)$ is reduced to the effective computation of $R_u(H')$. We expect that an elaboration of this method will be successful in extending the applicability of some recent algorithms developed by Minchenko, Ovchinnikov, and Singer to compute unipotent radicals for higher order equations.Comment: 12 page

Inverse decays and the relic density of the sterile sneutrino

We consider a weak scale supersymmetric seesaw model where the Higgsino is the next-to-lightest supersymmetric particle and the right-handed sneutrino is the dark matter candidate. It is shown that, in this model, inverse decays, which had been previously neglected, may suppress the sneutrino relic density by several orders of magnitude. After including such processes and numerically solving the appropriate Boltzmann equation, we study the dependence of the relic density on the mu parameter, the sneutrino mass, and the neutrino Yukawa coupling. We find that, even though much smaller than in earlier calculations, the sneutrino relic density is still larger than the observed dark matter density.Comment: 15 pages, 4 figure