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In a field, of course, ann-th degree polynomial has at mostn zeros. Is there anything like this in rings? We find: ann-th degree polynomial in a finite ringA, ofr elements, that is not zero everywhere inA is non-zero at leastr/2
times. This is sharp for alln, even in commutative rings; perhaps...
LetG be a group generated by a subset of elementsS. The Cayley diagram ofG givenS is the labeled directed graph with vertices identified with the elements ofG and (v, u) is an edge labeledh ifh ∈S anduh=v. The sequence of elements ofS corresponding to the edges transversed in a hamiltonian path...
Several characteristic functional equations satisfied by classes of polynomials of bounded degree are examined in connection with certain generalizations of the Morera-Carleman Theorem. Certain functional equations which have nonanalytic polynomial solutions are also considered.
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