منابع مشابه
SCHUR COVERS and CARLITZ’S CONJECTURE
We use the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz’s conjecture (1966). An exceptional polynomial f over a finite field Fq is a polynomial that is a permutation polynomial on infinitely many finite extensions of Fq. Carlitz’s conjecture says f must be of odd degree (if q is odd). Indeed, excluding characteristic 2 and 3, arithmetic ...
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Fix a rectangular Young diagram R, and consider all the products of Schur functions sλsλc , where λ and λ c run over all (unordered) pairs of partitions which are complementary with respect to R. Theorem: The self-complementary products, s λ where λ = λ, are linearly independent of all other sλsλc . Conjecture: The products sλsλc are all linearly independent.
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Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest possible number of consecutive quadratic non-residues modulo p is always less than p. (One can find a brief discussion of this problem in R. K. Guy’s book [5]). Schur also pointed out that the greatest number of consecutive q...
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In 1975 Wegner conjectured that the nerve of every finite good cover in R is d-collapsible. We disprove this conjecture. A good cover is a collection of open sets in R such that the intersection of every subcollection is either empty or homeomorphic to an open d-ball. A simplicial complex is d-collapsible if it can be reduced to an empty complex by repeatedly removing a face of dimension at mos...
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ژورنال
عنوان ژورنال: Israel Journal of Mathematics
سال: 1993
ISSN: 0021-2172,1565-8511
DOI: 10.1007/bf02808112