arXiv subjects

Jaydeep Chipalkatti

Publications and source records attributed to Jaydeep Chipalkatti.

25 records · Page 2Linked to original sources

arXiv2005

On the Wronskian combinants of binary forms

For generic binary forms $A_1,...,A_r$ of order $d$ we construct a class of combinants $C = \{\C_q: 0 \le q \le r, q \neq 1\}$, to be called the Wronskian combinants of the $A_i$. We show that the collection $C$ gives a projective imbedding of the Grassmannian $G(r,S_d)$, and as a corollary, any other combinant admits a formula as an iterated transvectant in the $C$. Our second main result characterizes those collections of binary forms which can arise as Wronskian combinants. These collections are the ones such that an associated algebraic differential equation has the maximal number of linearly independent polynomial solutions. Along the way we deduce some identities which connect Wronskians with transvectants.

math.AG↗

arXiv2005

Apolar schemes of algebraic forms

This is a note on the classical Waring's problem for several homogeneous forms. For positive integers (n,d,r,s), fix a general r-dimensional subspace of degree d forms in n+1 variables. We describe the family of s-sided polar polyhedra of this subspace in many specific cases.

math.AG↗

arXiv2005

The bipartite Brill-Gordan locus and angular momentum

This paper is a sequel to math.AG/0411110. Let P denote the projective space of degree d forms in n+1 variables. Let e denote an integer < d/2, and consider the subvariety X of forms which factor as L^{d-e} M^e for some linear forms L,M. In the language of our earlier paper, this is the Brill-Gordan locus associated to the partition (d-e,e). In this paper we calculate the Castelnuovo regularity of X precisely, and moreover show that X is r-normal for r at least 3. In the case of binary forms, we give a classical invariant-theoretic description of the defining equations of this locus in terms of covariants of d-ics. Modulo standard cohomological arguments, the proof crucially relies upon showing that certain 3j-symbols from the quantum theory of angular momentum are nonzero.

math.AG↗

arXiv2004

Brill-Gordan Loci, Transvectants and an Analogue of the Foulkes Conjecture

Combining a selection of tools from modern algebraic geometry, representation theory, the classical invariant theory of binary forms, together with explicit calculations with hypergeometric series and Feynman diagrams, we obtain the following interrelated results. A Castelnuovo-Mumford regularity bound and a projective normality result for the locus of hypersufaces that are equally supported on two hyperplanes. The surjectivity of an equivariant map between two plethystic compositions of symmetric powers; a statement which is reminiscent of the Foulkes-Howe conjecture. The nonvanishing of even transvectants of exact powers of generic binary forms. The nonvanishing of a collection of symmetric functions defined by sums over magic squares and transportation matrices with nonnegative integer entries. An explicit set of generators, in degree three, for the ideal of the coincident root locus of binary forms with only two roots of equal multiplicity.

math.AG↗

arXiv2004

A regularity result for a locus of Brill type

Let n,d be positive integers, with d even (say d=2e). Let X_(n,d) denote the locus of degree d hypersurfaces in P^n which consist of two e-fold hyperplanes. We bound the regularity of the ideal of this variety. Moreover, we show that this variety is r-normal for r at least 2. The proof of the latter part is is a result of a tripartite collaboration of algebraic geometry, classical invariant theory and theoretical physics. It is executed by reducing the question to a combinatorial calculation involving Feynman diagrams and hypergeometric functions.

math.AG↗

arXiv2002

The Waring loci of ternary quartics

Let F denote a homogeneous degree 4 polynomial in 3 variables, and let s be an integer between 1 and 5. We would like to know if F can be written as a sum of fourth powers of s linear forms (or a degeneration). We determine necessary and sufficient conditions for this to be possible. These conditions are expressed as the vanishing of certain concomitants of F for the natural action of SL_3.

math.AG↗

arXiv2001

On Waring's problem for several homogeneous forms

We reconsider the classical problem of representing a finite number of forms of degree d in n+1 variables as sums of powers of linear forms. We define a geometric construct called a `grove', which, in a number of cases allows us to determine the dimension of the space of forms which can be so represented. We also present two new `exceptional' examples, where this dimension is less than what a naive parameter count would predict.

math.AG↗