17,498 research outputs found
A generalization of the Subspace Theorem with polynomials of higher degree
Recently, Corvaja and Zannier obtained an extension of the Subspace Theorem
with arbitrary homogeneous polynomials of arbitrary degreee instead of linear
forms. Their result states that the set of solutions in P^n(K) (K number field)
of the inequality being considered is not Zariski dense. In our paper we prove
by a different method a generalization of their result, in which the solutions
are taken from an arbitrary projective variety X instead of P^n. Further, we
give a quantitative version which states in a precise form that the solutions
with large height lie ina finite number of proper subvarieties of X, with
explicit upper bounds for the number and for the degrees of these subvarieties.Comment: 31 page
Some generalizations of Schmidt's subspace theorem
The aim of this paper is twofold. The first is to give a quantitative version
of Schmidt's subspace theorem for arbitrary families of higher degree
polynomials. The second is to give a generalization of the subspace theorem for
arbitrary families of closed subschemes in algebraic projective varieties.Comment: In this version, some typos are corrected and the references list is
modified. The paper consists of 17 pages. arXiv admin note: substantial text
overlap with arXiv:1808.10286; text overlap with arXiv:math/0408381,
arXiv:1910.07966 by other author
Algebraic List-decoding of Subspace Codes
Subspace codes were introduced in order to correct errors and erasures for
randomized network coding, in the case where network topology is unknown (the
noncoherent case). Subspace codes are indeed collections of subspaces of a
certain vector space over a finite field. The Koetter-Kschischang construction
of subspace codes are similar to Reed-Solomon codes in that codewords are
obtained by evaluating certain (linearized) polynomials. In this paper, we
consider the problem of list-decoding the Koetter-Kschischang subspace codes.
In a sense, we are able to achieve for these codes what Sudan was able to
achieve for Reed-Solomon codes. In order to do so, we have to modify and
generalize the original Koetter-Kschischang construction in many important
respects. The end result is this: for any integer , our list- decoder
guarantees successful recovery of the message subspace provided that the
normalized dimension of the error is at most where
is the normalized packet rate. Just as in the case of Sudan's list-decoding
algorithm, this exceeds the previously best known error-correction radius
, demonstrated by Koetter and Kschischang, for low rates
Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials
The concept of cyclic tridiagonal pairs is introduced, and explicit examples
are given. For a fairly general class of cyclic tridiagonal pairs with
cyclicity N, we associate a pair of `divided polynomials'. The properties of
this pair generalize the ones of tridiagonal pairs of Racah type. The algebra
generated by the pair of divided polynomials is identified as a higher-order
generalization of the Onsager algebra. It can be viewed as a subalgebra of the
q-Onsager algebra for a proper specialization at q the primitive 2Nth root of
unity. Orthogonal polynomials beyond the Leonard duality are revisited in light
of this framework. In particular, certain second-order Dunkl shift operators
provide a realization of the divided polynomials at N=2 or q=i.Comment: 32 pages; v2: Appendices improved and extended, e.g. a proof of
irreducibility is added; v3: version for Linear Algebra and its Applications,
one assumption added in Appendix about eq. (A.2
Direction problems in affine spaces
This paper is a survey paper on old and recent results on direction problems
in finite dimensional affine spaces over a finite field.Comment: Academy Contact Forum "Galois geometries and applications", October
5, 2012, Brussels, Belgiu
Quasi-Exact Solvability and the direct approach to invariant subspaces
We propose a more direct approach to constructing differential operators that
preserve polynomial subspaces than the one based on considering elements of the
enveloping algebra of sl(2). This approach is used here to construct new
exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line
which are not Lie-algebraic. It is also applied to generate potentials with
multiple algebraic sectors. We discuss two illustrative examples of these two
applications: an interesting generalization of the Lam\'e potential which
posses four algebraic sectors, and a quasi-exactly solvable deformation of the
Morse potential which is not Lie-algebraic.Comment: 17 pages, 3 figure
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