1,720,973 research outputs found
The asymptotic leading term for maximum rank of ternary forms of a given degree
Get PDFLet 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(3,d)=d^2/4+O(d) as a consequence of the upper bound on rmax(3,d) given by the floor of (d^2+6d+1)/4
Sulle famiglie di curve con singolarità di prefissato tipo
No full textSi introduce la nozione di punto infinitamente vicino su uno schema noetheriano integro di dimensione due. Si dimostra che le curve di una serie lineare completa su una superficie liscia, con assegnato tipo debole di singolarità, costituiscono un sottoinsieme costruibile dello spazio proiettivo parametrizzante la serie completa stessa
A proof that the maximum rank for ternary quartics is seven
Get PDFAt the time of writing, the general problem of finding the maximum Waring rank for homogeneous polynomials of a fixed degree and in a fixed number of variables (or, equivalently, the maximum symmetric rank for symmetric tensors of a fixed order and in a 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 maximum rank for plane quartics is seven, following the elementary geometric idea of splitting power sum decompositions along three suitable lines
Review on book "Harmonic vector fields - Variational principles and differential geometry", by Dragomir, Sorin and Perrone, Domenico.
No full textOn Certain Classes of Curve Singularities with Reduced Tangent Cone
Get PDFWe 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
Differential invariants CoCoA sheets
No full textThis sofware was an essential tool for the results of a subsequent publication:
A. De Paris - A.M. Vinogradov, Scalar differential invariants of symplectic Monge–Ampère equations, Central European Journal of Mathematics, 9 (4), 731-751 (2011)
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On the injectivity of the nodal map
No full textWe address the problem of finding values of d, g, r such that the position of nodes of a plane projection (with only nodes) of a smooth curve in P^r of degree g and genus g does not uniquely determines the projection itself
On noncommutative equivariant bundles
Get PDFWe discuss a possible noncommutative generalization of the notion of an equivariant vector bundle. Let be a -algebra, a left -module, a Hopf -algebra, an algebra coaction, and let denote with the right -module structure induced by . The usual definitions of equivariant vector bundle naturally lead, in the context of -algebras, to an -module homomorphism [Theta: H otimes M to (H otimes A)_delta otimes_AM] that fulfills some appropriate conditions. On the other hand, sometimes an -Hopf module is considered instead, for the same purpose. When is invertible, as is always the case when 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 for which there exists such a that is not invertible and a left-right -Hopf module whose corresponding homomorphism is not an isomorphism
Reduced Tangent Cones and Conductor at Multiplanar Isolated Singularities
No full textWe prove a result on the associated graded ring of local algebras that satisfy some appropriate conditions. This allows us to compute, in the general case, the conductor of the local ring of singular isolated points on surfaces whose tangent cone is multiplanar; that is, it consists - as a set - of planes. We explicitly construct a class of such surfaces in the rational case
Some Formulae Arising in Projective-Differential Geometry
No full textWe set up a framework in which some projective-differential intuitive concepts can be very easily formalized. Then we prove some general formulas. Among them, two "inversion formulas" about deformations of families of nonreduced subschemes are particularly remarkable. As applications, we give convenient definitions about some classical topics, such as higher order foci; finally we propose a new proof of the classic structure theorem about the first Gauss map
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