Rational quartic spectrahedra in $3$-space are semialgebraic convex subsets $\mathbb{R} ^3$ of semidefinite, real symmetric $(4 \times 4)$-matrices, whose boundary admits a rational parameterization. The Zariski closure $\mathbb{C}\mathbb{P} the spectrahedron is complex symmetroid. We give necessary conditions on configurations singularities corresponding symmetroids \mathbb{P} spectrahedra. pr...

We give a class of exact solutions of quartic scalar field theories. These solutions prove to be interesting as are characterized by the production of mass contributions arising from the nonlinear terms while maintaining a wavelike behavior. So, a quartic massless equation has a nonlinear wave solution with a dispersion relation of a massive wave and a quartic scalar theory gets its mass term r...

Continuing the recent work of the second author, we prove that the diophantine equation fa(x, y) = x 4 − axy − xy + axy + y = 1 for |a| ≥ 3 has exactly 12 solutions except when |a| = 4, when it has 16 solutions. If α = α(a) denotes one of the zeros of fa(x, 1), then for |a| ≥ 4 we also find all γ ∈ Z[α] with Z[γ] = Z[α].

We compute the number of rational quartics on a general Calabi-Yau hypersurface in weighted projective space P(2, 1). The result agrees with the prediction made by mirror symmetry.

Dedicated to Robin Hartshorne in occasion of his 60th birthday.

We prove a general theorem that evaluates the Legendre symbol (A+B √ m p) under certain conditions on the integers A, B, m and the prime p. The evaluation is in terms of parameters appearing in a binary quadratic form representing p. The theorem has applications to quartic residuacity.

In this paper we give some results about primitive integral elements α in the family of bicyclic biquadratic fields Lc = Q( √

In this paper, based on techniques of Newton polygons, a result which allows the computation of a p integral basis of every quartic number field is given. For each prime integer p, this result allows to compute a p-integral basis of a quartic number field K defined by an irreducible polynomial P (X) = X4 + aX + b ∈ Z[X] in methodical and complete generality.

We review the recent results concerning the computation of cubic and quartic couplings of scalar fields in type IIB supergravity on AdS5 × S background that are dual to (extended) chiral primary operators in N = 4 SYM4. We discuss the vanishing of certain cubic and quartic couplings and non-renormalization property of corresponding correlators in the conformal field theory.