Searcharxiv⌕ Search

arXiv subjects

Alessandro De Paris

Publications and source records attributed to Alessandro De Paris.

11 recordsLinked to original sources
arXiv2025

Isomorphisms of graphs of Hyperbinary Expansions and Efficient Algorithms for Stern's Diatomic Sequence

To investigate hyperbinary expansions of a nonnegative integer~$n$, an edge-labeled directed graph $A(n)$ has recently been introduced. After pointing out some new simple facts about its cyclomatic number, we give a relatively simple description of its structure and prove that if $m,n$ are even numbers for which $A(n)$ and $A(m)$ are isomorphic as edge-labeled graphs, then $m=n$. From the structure of $A(n)$ we also derive a formula related to Stern's diatomic sequence, and in the same vein discuss some algorithms that recently appeared in the literature.

math.CO↗
arXiv2023

Structures of the Length Seven Power Sum Decompositions of Ternary Quartics

Motivated by the search for a deeper understanding of tensor rank, in view of its computational complexity applications, we investigate a possible path to determine the maximum symmetric rank in given degree and dimension. We work in terms of Waring rank of forms, and aiming to set up a firm basis for an induction procedure we examine some technical tools to organize length seven Waring decompositions of ternary quartics, that may turn to be fundamental.

math.AC↗
arXiv2018

On noncommutative equivariant bundles

We discuss a possible noncommutative generalization of the notion of an equivariant vector bundle. Let $A$ be a $\mathbb{K}$-algebra, $M$ a left $A$-module, $H$ a Hopf $\mathbb{K}$-algebra, $δ:A\to H\otimes A:=H\otimes_{\mathbb{K}} A$ an algebra coaction, and let $(H\otimes A)_δ$ denote $H\otimes A$ with the right $A$-module structure induced by~$δ$. The usual definitions of an equivariant vector bundle naturally lead, in the context of $\mathbb{K}$-algebras, to an $(H\otimes A)$-module homomorphism \[Θ:H\otimes M\to (H\otimes A)_δ\otimes_AM\] that fulfills some appropriate conditions. On the other hand, sometimes an $(A,H)$-Hopf module is considered instead, for the same purpose. When $Θ$ is invertible, as is always the case when $H$ is commutative, the two descriptions are equivalent. We point out that the two notions differ in general, by giving an example of a noncommutative Hopf algebra $H$ for which there exists such a $Θ$ that is not invertible and a left-right $(A,H)$-Hopf module whose corresponding homomorphism $M\otimes H\to (A\otimes H)_δ\otimes_AM$ is not an isomorphism.

math.RA↗
arXiv2018

Generic Power Sum Decompositions and Bounds for the Waring Rank

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 $d\ge 3$ can be annihilated by the product of $d-1$ pairwise independent linear forms.

math.AG↗
arXiv2017

High-rank ternary forms of even degree

We exhibit, for each even degree, a ternary form of rank strictly greater than the maximum rank of monomials. Together with an earlier result in the odd case, this gives the lower bound \[\operatorname{r_{max}}(3,d)\ge\left\lfloor\frac{d^2+2d+5}4\right\rfloor\] for $d\ge 2$, where $\operatorname{r_{max}}(n,d)$ denotes the maximum rank of degree $d$ forms in $n$ variables with coefficients in an algebraically closed field of characteristic zero.

math.AG↗
arXiv2015

The asymptotic leading term for maximum rank of ternary forms of a given degree

Let $\operatorname{r_{max}}(n,d)$ 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 $n,d\ge 3$, the value of $\operatorname{r_{max}}(n,d)$ is known only for $(n,d)=(3,3),(3,4),(3,5),(4,3)$. We prove that $\operatorname{r_{max}}(3,d)=d^2/4+O(d)$ as a consequence of the upper bound $\operatorname{r_{max}}(3,d)\le\left\lfloor\left(d^2+6d+1\right)/4\right\rfloor$.

math.AG↗
arXiv2015

Every Ternary Quintic is a Sum of Ten Fifth Powers

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 $d\le 4$. The best upper bound that is known for $d=5$ is twelve, and in this work we lower it to ten.

math.AG↗
arXiv2013

A Proof that the Maximal Rank for Plane Quartics is Seven

At the time of writing, the general problem of finding the maximal Waring rank for homogeneous polynomials of fixed degree and number of variables (or, equivalently, the maximal symmetric rank for symmetric tensors of fixed order and in fixed dimension) is still unsolved. To our knowledge, the answer for ternary quartics is not widely known and can only be found among the results of a master's thesis by Johannes Kleppe at the University of Oslo (1999). In the present work we give a (direct) proof that the maximal rank for plane quartics is seven, following the elementary geometric idea of splitting power sum decompositions along three suitable lines.

math.AG↗
arXiv2012

A remark on Waring decompositions of some special plane quartics

This work concerns Waring decompositions of a certain kind of plane quartics of high rank. The main result is the following. Let x, l_1, ...., l_7 be linear forms and q a quadratic form on a vector space of dimension 3. If x^2q=l_1^4+...+l_7^4 and the lines l_1=0, ..., l_7=0 in P^2 intersect x=0 in seven distinct points, then the line x=0 is (possibly improperly) tangent to the conic q=0.

math.AG↗
arXiv2011

Scalar Differential Invariants of Symplectic Monge-Ampère Equations

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 quasilinear equations of a suitable kind and in particular find a simple linearization criterion.

math.SG↗