Hasse Derivative Research Articles

On the convergence of series of fractional Hasse derivative bases in Fréchet spaces

This paper introduces a generalization of the Hasse derivative in the sense of the complex conformable derivative. The definition coincides with the classical version of the Hasse derivative of order . Accordingly, a new base, named complex conformable Hasse derivative bases (CCHDBs), is defined. We investigate the existence of expansions of analytic functions in a series of CCHDBs in Fréchet space on closed and open disks, open regions surrounding closed disks, for all entire functions and at the origin. Moreover, an upper bound for the order and type of the CCHDBs is obtained and proved to be attainable. The ‐property of CCHDBs is also discussed. Our results improve and extend the analog results in the complex analysis related to the classical complex derivative of a base of polynomials (BPs). The obtained results clarify several implications for the CCHDBs of special functions such as Euler, Bernoulli, Chebyshev, Bessel, and Gontcharaff polynomials.

Universality for Low-Degree Factors of Random Polynomials over Finite Fields

Abstract We show that the counts of low-degree irreducible factors of a random polynomial $f$ over $\mathbb {F}_q$ with independent but nonuniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for random polynomials over finite fields. Our strongest results require various assumptions on the parameters, but we are able to obtain results requiring only $q=p$ a prime with $p\leq \exp ({n^{1/13}})$ where $n$ is the degree of the polynomial. Our proofs use Fourier analysis and rely on tools recently applied by Breuillard and Varjú to study the $ax+b$ process, which show equidistribution for $f(\alpha )$ at a single point. We extend this to handle multiple roots and the Hasse derivatives of $f$, which allow us to study the irreducible factors with multiplicity.

Zeros with multiplicity, Hasse derivatives and linear factors of general skew polynomials

In this work, multiplicities of zeros of skew polynomials are studied. Two distinct definitions are considered: First, a is said to be a zero of F of multiplicity r if divides F on the right; second, a is said to be a zero of F of multiplicity r if some skew polynomial having as its only right zero, divides F on the right. Neither of these two notions implies the other for general skew polynomials. We show that, in the first case, Lam and Leroy’s concept of P-independence does not behave naturally, whereas a union theorem still holds. In contrast, we show that P-independence behaves naturally for the second notion of multiplicities. As a consequence, we provide extensions of classical commutative results to general skew polynomials. These include: (1) The upper bound on the number of (P-independent) zeros (counting multiplicities) of a skew polynomial by its degree, and (2) The equivalence of P-independence, Hermite interpolation and the invertibility of confluent Vandermonde matrices (for which we introduce skew polynomial Hasse derivatives).

A general construction of Reed-Solomon codes based on generalized discrete Fourier transform

In this paper, we employ the concept of the Generalized Discrete Fourier Transform, which in turn relies on the Hasse derivative of polynomials, to give a general construction of Reed-Solomon codes over Galois fields of characteristic not necessarily co-prime with the length of the code. The constructed linear codes enjoy nice algebraic properties just as the classic one.

Bounding the Number of Common Zeros of Multivariate Polynomials and Their Consecutive Derivatives

We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

Two-dimensional generalized discrete Fourier transform and related quasi-cyclic Reed--Solomon codes

Using the concept of the partial Hasse derivative, we introduce a generalization of the classical 2-dimensional discrete Fourier transform, which will be called 2D-GDFT. Begining with the basic properties of 2D-GDFT, we proceed to study its computational aspects as well as the inverse transform, which necessitate the development of a faster way to calculate the 2D-GDFT. As an application, we will employ 2D-GDFT to construct a new family of quasi-cyclic linear codes that can be assumed to be a generalization of Reed‒Solomon codes.

More results on the number of zeros of multiplicity at least [formula omitted

We consider multivariate polynomials and investigate how many zeros of multiplicity at least r they can have over a Cartesian product of finite subsets of a field. Here r is any prescribed positive integer and the definition of multiplicity that we use is the one related to Hasse derivatives. As a generalization of material in Augot and Stepanov (2009) and Dvir et al. (2009) a general version of the Schwartz–Zippel was presented in Geil and Thomsen (2013) which from the leading monomial – with respect to a lexicographic ordering – estimates the sum of zeros when counted with multiplicity. The corresponding corollary on the number of zeros of multiplicity at least r is in general not sharp and therefore in Geil and Thomsen (2013) a recursively defined function D was introduced using which one can derive improved information. The recursive function being rather complicated, the only known closed formula consequences of it are for the case of two variables (Geil and Thomsen, 2013). In the present paper we derive closed formula consequences for arbitrary many variables, but for the powers in the leading monomial being not too large. Our bound can be viewed as a generalization of the footprint bound (Høholdt, 1998; Geil and Høholdt, 2000)—the classical footprint bound taking not multiplicity into account.

BCH Codes for the Rosenbloom–Tsfasman Metric

The Rosenbloom–Tsfasman metric has attracted the attention of many researchers as a generalization of the Hamming metric that is relevant to practical problems. Codes for this metric were considered. In particular, Reed–Solomon codes were generalized to be compatible with this metric. In this paper, a generalization of BCH codes for the Rosenbloom–Tsfasman metric is proposed. This generalization is based on considering BCH codes as subfield subcodes of Reed–Solomon codes. By characterizing these subfield subcodes, an explicit construction of BCH codes for the Rosenbloom–Tsfasman metric is provided. Two important properties of Reed–Solomon codes and BCH codes for the Rosenbloom–Tsfasman metric are studied and compared with those for the Hamming metric. These properties are cyclic structure and duality. The approach is based on Galois-Fourier transforms associated with Hasse derivatives.

Prefactor Reduction of the Guruswami–Sudan Interpolation Step

The most computationally intensive step of the Guruswami–Sudan list decoder for generalized Reed–Solomon codes is the formation of a bivariate interpolation polynomial. Complexity can be reduced if this polynomial has prefactors, i.e., factors of its univariate constituent polynomials that are independent of the received vector, and hence known a priori . For example, the well-known re-encoding projection due to Koetter et al. leads to one class of prefactors. This paper introduces so-called Sierpinski prefactors that result from the property that many binomial coefficients, which arise in the multiplicity constraints defined in terms of the Hasse derivative, are zero modulo the underlying field characteristic. It is shown that re-encoding prefactors and Sierpinski prefactors can be combined to achieve a significantly reduced Guruswami–Sudan interpolation step. In certain practically relevant cases, the introduction of Sierpinski prefactors reduces the number of unknown polynomial coefficients by an additional 10% or more (beyond the reduction due to re-encoding prefactors alone), without incurring additional computational effort at the decoder.

On the Algebraic Structure of Quasi-cyclic Codes IV: Repeated Roots

A trace formula for quasi-cyclic codes over rings of characteristic not coprime with the co-index is derived. The main working tool is the Generalized Discrete Fourier Transform (GDFT), which in turn relies on the Hasse derivative of polynomials. A characterization of Type II self-dual quasi-cyclic codes of singly even co-index over finite fields of even characteristic follows. Implications for generator theory are shown. Explicit expressions for the combinatorial duocubic, duoquintic and duoseptic constructions in characteristic two over finite fields are given.

Towards a VLSI Architecture for Interpolation-Based Soft-Decision Reed-Solomon Decoders

The Koetter-Vardy algorithm is an algebraic soft-decision decoder for Reed-Solomon codes which is based on the Guruswami-Sudan list decoder. There are three main steps: (1) multiplicity calculation, (2) interpolation and (3) root finding. The Koetter-Vardy algorithm seems challenging to implement due to the high cost of interpolation. Motivated by a VLSI implementation viewpoint we propose an improvement to the interpolation algorithm that uses a transformation of the received word to reduce the number of iterations. We show how to reduce the memory requirements and give an efficient VLSI implementation for the Hasse derivative.

Subresultants and locally nilpotent derivations

In this paper we establish a connection between subresultants and locally nilpotent derivations over commutative rings containing the rationals. As consequence of this connection, we prove that for any commutative ring with unit and any polynomials P and Q in A[y] , the ith subresultant of P and Q is the determinant of a matrix, depending only on the degrees of P and Q, whose entries are taken from the list built with P, Q and their successive Hasse derivatives.

On the Jordan form of a family of linear mappings

Let A and B be square matrices over a field F having their eigenvalues λ and μ in F, and let g( x, y) = ∑ r, s a rs x r y s be a polynomial over F. Assuming the Jordan forms of A and B to be known, the Jordan form of ∑ r, s a rs A r ⊗ B s is determined when the partial derivatives ∂g ∂x and ∂g ∂y are nonzero at (λ, μ), thus generalizing the classical cases g( x, y) = x + y and g( x, y) = xy. The particular cases g( x, y) = x k + y l and g( x, y) = x k y l are also generalized by using properties of the partial Hasse derivatives of g at (λ, μ). The case where A and B are 2 × 2 or 3 × 3 matrices, g being arbitrary, is discussed exhaustively.

On repeated-root cyclic codes

A parity-check matrix for a q-ary repeated-root cyclic code is derived using the Hasse derivative. Then the minimum distance of a q-ary repeated-root cyclic code is expressed in terms of the minimum distance of a certain simple-root cyclic code. With the help of this result, several binary repeated-root cyclic codes of lengths up to n=62 are shown to contain the largest known number of codewords for their given length and minimum distance. The relative minimum distance d/sub min//n of q-ary repeated-root cyclic codes of rate r>or=R is proven to tend to zero as the largest multiplicity of a root of the generator g(x) increases to infinity. It is further shown that repeated-root cycle codes cannot be asymptotically better than simple-root cyclic codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>