Abstract. Let rmax(n, d) be the maximum Waring rank for the set of all homogeneous polynomials of degree d > 0 in n indeterminates with coefficients in an algebraically closed field of characteristic zero. To our knowledge, when n, d ≥ 3, the value of rmax(n, d) is known only for (n, d) = (3, 3), (3, 4), (3, 5), (4, 3). We prove that rmax (

A notion of open rank, related with generic power sum decompositions of forms, has recently been introduced in the literature. The main result here is that the maximum open rank for plane quartics is eight. In particular, this gives the first example of n, d, such that the maximum open rank for degree d forms that essentially depend on n variables is strictly greater than the maximum rank. On one hand, the result allows to improve the previously known bounds on open rank, but on the other hand indicates that such bounds are likely quite relaxed. Nevertheless, some of the preparatory results are of independent interest, and still may provide useful information in connection with the problem of finding the maximum rank for the set of all forms of given degree and number of variables. For instance, we get that every ternary forms of degree d ≥ 3 can be annihilated by the product of d − 1 pairwise independent linear forms.

The response of olfactory receptor neurons to odor mixtures is not well understood. Here, using experimental constraints, we investigate the mathematical structure of the odor response space and its consequences. The analysis suggests that the odor response space is 3-dimensional, and predicts that the dose-response curve of an odor receptor can be obtained, in most cases, from three primary components with specific properties. This opens the way to an objective procedure to obtain specific olfactory receptor responses by manipulating mixtures in a mathematically predictable manner. This result is general and applies, independently of the number of odor components, to any olfactory sensory neuron type with a response curve that can be represented as a sigmoidal function of the odor concentration.

Abstract. To our knowledge at the time of writing, the maximum Waring rank for the set of all ternary forms of degree d (with coefficients in an algebraically closed field of characteristic zero) is known only for d ≤ 4. The best upper bound that is known for d = 5 is twelve, and in this work we lower it to ten.

Abstract. All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce a series of invariant differential forms and vector fields which allows us to construct numerous scalar differential invariants of higher order. The introduced invariants give a solution of the symplectic equivalence of Monge-Ampère equations. As an example we study equations of the form uxy + f (x, y, ux, uy) = 0 and in particular find a simple linearization criterion.