Monomials as sums of powers: The real binary case

Aequationes mathematicae

In this paper we consider a generalized monomial or polynomial $$ f : \mathbb {R}\rightarrow \mathbb {R}$$ f : R → R that satisfies the additional equation $$ f(x) f(y) = 0 $$ f ( x ) f ( y ) = 0 for the pairs $$ (x,y) \in D \,$$ ( x , y ) ∈ D , where $$ D \subseteq {\mathbb {R}}^{2} $$ D ⊆ R 2 is given by some algebraic condition. In the particular cases when f is a generalized polynomial and there exist non-constant regular polynomials p and q that fulfill $$\begin{aligned} D = \{\, (p(t),q(t)) \,\vert \, t \in \mathbb {R}\,\} \end{aligned}$$ D = { ( p ( t ) , q ( t ) ) | t ∈ R } or f is a generalized monomial and there exists a positive rational m fulfilling $$\begin{aligned} D = \{\, (x,y) \in {\mathbb {R}}^{2} \,\vert \, x^2 - m y^2 = 1 \,\}, \end{aligned}$$ D = { ( x , y ) ∈ R 2 | x 2 - m y 2 = 1 } , we prove that $$ f(x) = 0 $$ f ( x ) = 0 for all $$ x \in \mathbb {R}\,$$ x ∈ R .

2021

This paper concerns the exponentiation of monomial ideals. While it is customary for the exponentiation operation on ideals to consider natural powers, we extend this notion to powers where the exponent is a positive real number. Real powers of a monomial ideal generalize the integral closure operation and highlight many interesting connections to the theory of convex polytopes. We provide multiple algorithms for computing the real powers of a monomial ideal. An important result is that given any monomial ideal I, the function taking real numbers to the corresponding real power of I is a step function whose jumping points are rational. This reduces the problem of determining real powers to rational exponents.

1998

The variety of sums of powers of a homogeneous polynomial of degree d in n variables is defined and investigated in some examples, old and new. These varieties are studied via apolarity and syzygies. Classical results (cf. [Sylvester 1851], [Hilbert 1888], [Dixon, Stuart 1906]) and some more recent results of Mukai (cf. [Mukai 1992]) are presented together with new results for the cases (n, d) = (3, 8), (4, 2), (5, 3).

2007

The monomiality principle was introduced (see [1] and the references therein) in order to derive the properties of special or generalized polynomials starting from the corresponding ones of monomials. We show a general technique of extending the monomiality approach to multi-index polynomials in several variables. Application of this technique to the case of Hermite, Laguerre-type and mixed-type (i.e., between Laguerre and Hermite) polynomials is given.

arXiv (Cornell University), 2021

This paper concerns fractional powers of monomial ideals. Rational powers of a monomial ideal generalize the integral closure operation as well as recover the family of symbolic powers. They also highlight many interesting connections to the theory of convex polytopes. We provide multiple algorithms for computing the rational powers of a monomial ideal. We also introduce a mild generalization allowing real powers of monomial ideals. An important result is that given any monomial ideal I, the function taking a real number to the corresponding real power of I is a step function which is left continuous and has rational discontinuity points.

Proceedings of the American Mathematical Society, 2017

We give a criterion which characterizes a homogeneous real multi-variate polynomial to have the property that all sufficiently large powers of the polynomial (as well as their products with any given positive homogeneous polynomial) have all positive coefficients. Our result generalizes a result of De Angelis, which corresponds to the case of homogeneous bivariate polynomials, as well as a classical result of Pólya, which corresponds to the case of a specific linear polynomial. As an application, we also give a characterization of certain polynomial spectral radius functions of the defining matrix functions of Markov chains.

Journal of Pure and Applied Algebra, 2021

Let f be the F q-linear map over F q 2n defined by x → x + ax q s + bx q n+s with gcd(n, s) = 1. It is known that the kernel of f has dimension at most 2, as proved by Csajbók et al. in "A new family of MRD-codes" (2018). For n big enough, e.g. n ≥ 5 when s = 1, we classify the values of b/a such that the kernel of f has dimension at most 1. To this aim, we translate the problem into the study of some algebraic curves of small degree with respect to the degree of f ; this allows to use intersection theory and function field theory together with the Hasse-Weil bound. Our result implies a non-scatteredness result for certain high degree scattered binomials, and the asymptotic classification of a family of rank metric codes.

Singularities — Kagoshima 2017

International Journal of Number Theory, 2017

We study Laurent polynomials in any number of variables that are sums of at most [Formula: see text] monomials. We first show that the Mahler measure of such a polynomial is at least [Formula: see text], where [Formula: see text] is the height of the polynomial. Next, restricting to such polynomials having integer coefficients, we show that the set of logarithmic Mahler measures of the elements of this restricted set is a closed subset of the nonnegative real line, with [Formula: see text] being an isolated point of the set. In the final section, we discuss the extent to which such an integer polynomial of Mahler measure [Formula: see text] is determined by its [Formula: see text] coefficients.