Higher Level Completeness for Permutation Polynomials

• URL: http://arxiv.org/abs/2310.12466v1
• Date: Thu, 19 Oct 2023 04:47:53 GMT
• Title: Higher Level Completeness for Permutation Polynomials
• Authors: S. Rajagopal, P. Vanchinathan,
• Abstract summary: Generalising the concept of a complete permutation over a finite field, we define completness to level $kge1$ in fields of odd characteristics. We construct two families of characteristics that satisfy the condition of high level completeness for all finite fields.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Generalising the concept of a complete permutation polynomial over a finite field, we define completness to level $k$ for $k\ge1$ in fields of odd characteristic. We construct two families of polynomials that satisfy the condition of high level completeness for all finite fields, and two more families complete to the maximum level a possible for large collection of finite fields. Under the binary operation of composition of functions one family of polynomials is an abelian group isomorphic to the additive group, while the other is isomorphic to the multiplicative group.

Related papers

• On Maximal Families of Binary Polynomials with Pairwise Linear Common Factors [1.249418440326334]
  We characterize maximal families over the binary field $mathbbF$. Our findings prompt several more open questions, which we plan to address in an extended version of this work.
  arXiv  Detail & Related papers  (2024-05-14T16:30:28Z)
• Structure of universal formulas [13.794391803767617]
  We introduce a hierarchy of classes connecting the global approximability property to the weaker property of infinite VC dimension. We show that fixed-size neural networks with not more than one layer of neurons having activations cannot approximate functions on arbitrary finite sets. We give examples of functional families, including two-hidden-layer neural networks, that approximate functions on arbitrary finite sets, but fail to do that on the whole domain of definition.
  arXiv  Detail & Related papers  (2023-11-07T11:50:25Z)
• Two-body Coulomb problem and hidden $g^{(2)}$ algebra: superintegrability and cubic polynomial algebra [55.2480439325792]
  It is shown that the two-body Coulomb problem in the Sturm representation leads to a new two-dimensional, exactly-solvable, superintegrable quantum system in curved space. The two integrals are integral of orders two and four, they are made from two components of the angular momentum and from the modified LaplaceRunge-Lenz vector.
  arXiv  Detail & Related papers  (2023-09-28T22:47:18Z)
• An Analysis of On-the-fly Determinization of Finite-state Automata [65.268245109828]
  We establish an abstraction of on-the-fly determinization of finite-state automata and demonstrate how it can be applied to bound the automatons. A special case of our findings is that automata with many non-deterministic transitions almost always admit a determinization of complexity.
  arXiv  Detail & Related papers  (2023-08-27T11:51:27Z)
• An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree [79.43134049617873]
  In this paper, we demonstrate an exponential separation between exact degree and approximate quantum query for a partial function. For an alphabet size, we have a constant versus separation complexity.
  arXiv  Detail & Related papers  (2023-01-22T22:08:28Z)
• Grothendieck inequalities characterize converses to the polynomial method [1.137457877869062]
  A surprising 'converse to the method' of Aaronson et al. (CCC16) shows that any bounded quadratic can be computed exactly by a 1-query up to a universal multiplicative factor related to the Grothendieck constant.
  arXiv  Detail & Related papers  (2022-12-16T16:26:04Z)
• On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials [0.0]
  This work reports and classifies the most general construction of rational quantum potentials in terms of the generalized Hermites. It is achieved by exploiting the intrinsic relation between third-order shape-invariant Hamiltonians and the fourth Painlev'e equation.
  arXiv  Detail & Related papers  (2022-03-10T20:45:37Z)
• Progressive approximation of bound states by finite series of square-integrable functions [0.0]
  We use the "tridiagonal representation approach" to solve the time-independent Schr"odinger equation for bound states in a basis set of finite size.
  arXiv  Detail & Related papers  (2022-02-20T00:25:35Z)
• Abelian Neural Networks [48.52497085313911]
  We first construct a neural network architecture for Abelian group operations and derive a universal approximation property. We extend it to Abelian semigroup operations using the characterization of associative symmetrics. We train our models over fixed word embeddings and demonstrate improved performance over the original word2vec.
  arXiv  Detail & Related papers  (2021-02-24T11:52:21Z)
• Finite-Function-Encoding Quantum States [52.77024349608834]
  We introduce finite-function-encoding (FFE) states which encode arbitrary $d$-valued logic functions. We investigate some of their structural properties.
  arXiv  Detail & Related papers  (2020-12-01T13:53:23Z)
• A refinement of Reznick's Positivstellensatz with applications to quantum information theory [72.8349503901712]
  In Hilbert's 17th problem Artin showed that any positive definite in several variables can be written as the quotient of two sums of squares. Reznick showed that the denominator in Artin's result can always be chosen as an $N$-th power of the squared norm of the variables.
  arXiv  Detail & Related papers  (2019-09-04T11:46:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.