On Certain Classes of Curve Singularities with Reduced Tangent Cone

We study a class of rational curves with an ordinary singular point, which was introduced in [Geramita and Orecchia, Minimally Generating Ideals Defining Certain Tangent Cones, J. of Algebra 78, No. 1 (1982), 36 – 57]. We find some conditions under which the tangent cone is reduced and we show that the tangent cone is not always reduced. We construct another class of rational curves with an ordinary singular point satisfying the condition required in [Ibid.] and whose tangent cone is always reduced

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 $\operatorname{r_{max}}(3,d)\le\left\lfloor\left(d^2+6d+1\right)/4\right\rfloor$.Comment: v1: 10 pages. v2: extended introduction and some mistakes correcte

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

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.Comment: Relevant information added in the footnote (1) at p.

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, $\delta:A\to H\otimes A:=H\otimes_{\mathbb{K}} A$ an algebra coaction, and let $(H\otimes A)_\delta$ denote $H\otimes A$ with the right $A$-module structure induced by~$\delta$. The usual definitions of an equivariant vector bundle naturally lead, in the context of $\mathbb{K}$-algebras, to an $(H\otimes A)$-module homomorphism $$\Theta:H\otimes M\to (H\otimes A)_\delta\otimes_AM$$ that fulfills some appropriate conditions. On the other hand, sometimes an $(A,H)$-Hopf module is considered instead, for the same purpose. When $\Theta$ 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 $\Theta$ that is not invertible and a left-right $(A,H)$-Hopf module whose corresponding homomorphism $M\otimes H\to (A\otimes H)_\delta\otimes_AM$ is not an isomorphism.Comment: In this version we dismiss the term neb-homomorphism (hinting at 'noncommutative equivariant bundles'), as the class of modules is larger than the class of algebraic counterparts of vector bundles. We also corrected some mistakes. Our main example does not immediately extended to the left-right case and the example about the 'exotic' Hopf module works only in the left-right cas

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.Comment: Accepted version. The final publication is available at link.springer.com: http://link.springer.com/article/10.1007/s00454-017-9886-

Scalar differential invariants of symplectic Monge–Ampère equations

All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère PDEs 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. A series of invariant differential forms and vector fields are also introduced: they allow one to construct numerous scalar differential invariants of higher order. The introduced invariants give a solution to the symplectic equivalence problem for Monge-Ampère equations

Corrigendum to "Reduced Tangent Cones and Conductor at Multiplanar Isolated Singularities"

We explicitly fix a mistake in a preliminary statement of our previous paper on the conductor at a multiplanar singularity. The correction is not immediate and, though the mistake does not affect correctness of the subsequent results, the wrong statement could easily be misleading.Comment: Corrigendum to a published pape

Quantum limits to estimation of photon deformation

We address potential deviations of radiation field from the bosonic behaviour and employ local quantum estimation theory to evaluate the ultimate bounds to precision in the estimation of these deviations using quantum-limited measurements on optical signals. We consider different classes of boson deformation and found that intensity measurement on coherent or thermal states would be suitable for their detection making, at least in principle, tests of boson deformation feasible with current quantum optical technology. On the other hand, we found that the quantum signal-to-noise ratio (QSNR) is vanishing with the deformation itself for all the considered classes of deformations and probe signals, thus making any estimation procedure of photon deformation inherently inefficient. A partial way out is provided by the polynomial dependence of the QSNR on the average number of photon, which suggests that, in principle, it would be possible to detect deformation by intensity measurements on high-energy thermal states.Comment: 9 page

Quantum BRST operators in the extended BRST-anti-BRST formalism

The quantum BRST-anti-BRST operators are explicitely derived and the consequences related to correlation functions are investigated. The connection with the standard formalism and the loopwise expansions for quantum operators and anomalies in Sp(2) approach are analyzed.Comment: 7 page