A note on the minimal polynomial of the Kronecker sum of two linear operators

Silva, J.A. Dias da; Hamidoune, Yahya Ould

doi:10.1016/0024-3795(90)90326-8

Cited by 14 publications

(7 citation statements)

References 3 publications

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“…On the other hand, if A is an arithmetic progression then, for each j, 0 j A is also an arithmetic progression with the same difference. Therefore, using the results of [7,8], we conclude that if A is an arithmetic progression then equality holds in (4).…”

Section: Lower Bound For the Cardinality Of^m K Amentioning

confidence: 79%

“…By " we represent the alternating character of S m over F. We consider the elements of the group algebra F½S m defined by Let V and W be vector spaces over F. Let v 2 V, w 2 W, f be a linear operator on V and g be a linear operator on W. In [7] it was proved that the dimension of the cyclic subspace of v w associated with the Kronecker sum of f and g is bounded bellow either by the characteristic of the field F or by the sum of the dimensions of the cyclic subspaces of v and w minus one, i.e., dim C f IþIg ðv wÞ ! min p, dim C f ðvÞ þ dim C g ðwÞ À 1 È É :…”

Section: Linear Algebraic Additive Resultsmentioning

confidence: 99%

“…Some problems of Additive Number Theory have been solved by using a linear algebraic approach [7,8]. This approach is based on the fact that for a simple structure linear operator the degree of the minimal polynomial is equal to the cardinality of its spectrum.…”

Section: Introductionmentioning

confidence: 99%

“…This lower bound holds for an arbitrary field [7]. A problem related with this one was considered by Erdo¨s and Heilbronn and concerns the study of lower bounds for the cardinality of the set of restricted sums of pairs of elements of a subset A of Z p .…”

Section: Introductionmentioning

confidence: 99%

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Young symmetrizers and additive theory

Caldeira

Silva

2003

Linear and Multilinear Algebra

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For partition of m and A finite nonempty subset of a field we define the set of -restricted sums of m-tuples of elements of A,^m A, and using Additive Number Theory results from [J.A. Dias da Silva and Y.O. Hamidoune (1990). A note on the minimal polynomial of the Kronecker sum of two linear operators. Linear Algebra Appl., 141, 283-287; J.A. Dias da Silva and Y.O. Hamidoune (1994). Cyclic spaces for Grassmann derivatives and additive theory. Bull. London Math. Soc., 26, 140-146] we obtain a lower bound for its cardinality. Next, using results and techniques from [J.A. Dias da Silva and Y.O. Hamidoune (1990). A note on the minimal polynomial of the Kronecker sum of two linear operators. Linear Algebra Appl., 141, 283-287; J.A. Dias da Silva and Y.O. Hamidoune (1994). Cyclic spaces for Grassmann derivatives and additive theory. Bull. London Math. Soc., 26, 140-146] we obtain lower bounds for the degrees of minimal polynomials of restrictions of derivations to ranges of Young symmetrizers and to the symmetry class of tensors V , and we show that the lower bound for the cardinality of^m A can also be obtained from these lower bounds.

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Section: Lower Bound For the Cardinality Of^m K Amentioning

confidence: 79%

Section: Linear Algebraic Additive Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Young symmetrizers and additive theory

Caldeira

Silva

2003

Linear and Multilinear Algebra

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“…First, we state three well known theorems: the first one is a generalized version of what is known as the Caveman Theorem [12], followed by a generalized form of the Erdős-Ginzburg-Ziv (EGZ) Theorem [1], [11], and the affirmed EHC [2], [9] …”

Section: Preliminariesmentioning

confidence: 99%

On some developments of the Erdős–Ginzburg–Ziv Theorem II

et al. 2003

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1. Introduction. Let S be a sequence of elements from the cyclic group Z m . We say S is zsf (zero-sum free) if there does not exist an m-term subsequence of S whose sum is zero. Let g(m, k) (resp. g * (m, k)) denote the least integer such that every sequence S with at least (resp. with exactly) k distinct elements and length g(m, k) (resp. g * (m, k)) must contain an m-term subsequence whose sum is zero. By an affine transformation in Z m we mean a map of the form x → ax + b, with a, b ∈ Z m and gcd(a, m) = 1. Furthermore, let E(m, s) denote the set of all equivalence classes of zsf sequences S of length s, up to order and affine transformation, that are not a proper subsequence of another zsf sequence. Using the above notation, the renowned Erdős-Ginzburg-Ziv Theorem (The function g(m, k) was introduced in [4], where it was shown that g(m, 4) = 2m − 3 for m ≥ 4. Furthermore, based on a lower bound construction the authors conjectured the value of g(m, k) for fixed k and sufficiently large m. Concerning the upper bound, they established an upper bound for m prime modulo the affirmation of the Erdős-Heilbronn conjecture (EHC). Since then, the EHC has been affirmed [9], [2], moreover, the bound given in [4] was extended for nonprimes in [19]. As will later be seen, it is worthwhile to mention that the affirmation of the EHC has resulted in several attempted generalizations and related results [6]

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Ballot Numbers, Alternating Products, and the Erdős-Heilbronn Conjecture

Nathanson

2013

The Mathematics of Paul Erdős I

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A note on the minimal polynomial of the Kronecker sum of two linear operators

Cited by 14 publications

References 3 publications

Young symmetrizers and additive theory

Young symmetrizers and additive theory

On some developments of the Erdős–Ginzburg–Ziv Theorem II

Ballot Numbers, Alternating Products, and the Erdős-Heilbronn Conjecture

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