Related papers: On scalar-valued nonlinear absolutely summing mapp…

In this note we consider algebraic exponential sums over the values of homogeneous nonsingular polynomials $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ in the quotient ring $\mathbb{Z}/p^2\mathbb{Z}$. We provide an estimate of…

In this paper, we derive some interesting symmetric properties for the geenralized Euler numbers and polynomials.

We introduce a general definition of almost $p$-summing mappings and give several concrete examples of such mappings. Some known results are considerably generalized and we present various situations in which the space of almost $p$-summing…

We consider two seemingly unrelated questions: the relationship between nonnegative polynomials and sums of squares on real varieties, and sparse semidefinite programming. This connection is natural when a real variety $X$ is defined by a…

Let $F$ be an algebraically closed field of characteristic different from $2$. We show that the images of multilinear $*$-polynomials on $UT_2$ are homogeneous vector spaces. An analogous result holds for $UT_3$ endowed with non-trivial…

In the present paper a new mean value theorem for polynomials of special form is obtained. The case of sums on vertices of a regular polygon is studied. A criterion for a certain equation to be satisfied is obtained.

We provide upper bounds for the cardinality of the value set of a polynomial map in several variables over a finite field. These bounds generalize earlier bounds for univariate polynomials.

In this paper, we develop the theory of absolutely summing multipolynomials. Among other results, we generalize and unify previous works of G. Botelho and D. Pellegrino concerning absolutely summing polynomials/multilinear mappings in…

In the smallest cases where there exist nonnegative polynomials that are not sums of squares we present a complete explanation of this distinction. The fundamental reason that the cone of sums of squares is strictly contained in the cone of…

We discuss various aspects of representation of a polynomial as a sum of monomials (for example, uniqueness of such representation and related estimations).

We propose investigating a summation analog of the paradigm for parallel integration. We make some first steps towards an indefinite summation method applicable to summands that rationally depend on the summation index and a P-recursive…

We study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in $n$ variables. The main tool is combinatorial polarization, and the…

We consider the problem of extending the classical S-lemma from commutative case to noncommutative cases. We show that a symmetric quadratic homogeneous matrix-valued polynomial is positive semidefinite if and only if its coefficient matrix…

In these introductory lectures we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and massive Feynman diagrams at higher loops.…

We present a concentration inequality for linear functionals of noncommutative polynomials in random matrices. Our hypotheses cover most standard ensembles, including Gaussian matrices, matrices with independent uniformly bounded entries…

In this note we prove that certain polynomial mappings $P_\mathfrak{g}^k(\mathbf{x}) \in \mathbf{Z}[\mathbf{x}]$ in $n$-variables obtained from simple complex Lie algebras $\mathfrak{g}$ of arbitrary rank $n \ge 1$, are exceptional.

We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. In particular, we show that chaotic random…

We show that any two holomorhpic maps, not both of which are constant, from a generalized Hopf manifold to its base must have a coincidence. We prove a similar result for holomorphic maps from a generalized Calabi-Eckmann manifold.

The present note considers a certain family of sums indexed by the set of fixed length compositions of a given number. The sums in question cannot be realized as weighted compositions. However they can be be related to the hypergeometric…

These lecture notes provide an informal introduction to the theory of nonnegative polynomials and sums of squares. We highlight the history and some recent developments, especially the new connections with classical (complex) algebraic…