Antonio CAUSA

My research interests mainly concern Algebraic Geometry, Coding Theory, Variational Inequalities.
Some results have been obtained on the "Waring problem" for real binary symmetric tensors.
The Waring problem for symmetric tensors deals with the following problem: given a degree n polynomial in m variables find the (Waring) rank of f, i.e. the minimum number of summands that achieve the following equality: f=a₁l₁ⁿ +...+ a_kl_kⁿ with a_i real coefficients and l_i linear forms in m variables.
In the context of Variational Inequalities it has been studied the regularity of the solution of some classes of variational inequalities.
Let K be a nonempty, closed and convex subset of the n-dimensional Euclidean space Rⁿ, F : U → Rⁿ a continuous mapping. The variational inequality problem (VI) is the problem of finding a point x* in K such that
F(x*)(y − x*) ≥ 0 ∀y in K.
If the set K and the mappings F are dependent by a (usually real) parameter t one can ask which properties solution x*(t) fulfills.
When the convex set K(t) is a polytope, i.e. a bounded intersection of a finite set of half-spaces, it has been studied Lipschitz continuity of solution x*(t).
During the last decades  many scholars have been studying the connections between commutative algebra and combinatorics. It has been noted that problems in the former can be translated in  the language of the latter and vice versa.
As a starting point some combinatorial objects as graphs and hypergraphs can be used to construct particular monomial ideals called Edge ideals and Covering ideals. In this context we are interested in studying some algebraic invariants such Betti numbers, Hilbert function, Castelnuovo-Mumford regularity, Waldschmidt constant of Edge ideals and Covering ideals associated to matroids or Steiner systems.
A Matroid is a hypergraph which consists of a a finite set M of elements together with a family of subsets of M, called independent sets, such that the empty set is independent, every subset of an independent set is independent, for every subset A of M, all maximum independent sets contained in A have the same number of elements.
Another interesting problem is the Ideal containment problem for Edge ideals and Covering ideals, i.e. decide for  which m and r there is the containment I^(m)  is a subset of I^r, where I^(m) is the symbolic power of the ideal I and I^r is the regular power of I.