| dc.description.abstract |
Additive bases, and less importantly multiplicative bases, have been ex-
tensively studied for several centuries. More recently, expanding polynomi-
als (of course, with more than one variable) have been considered with a
view to studying the expansion of nite sets under these polynomials. If
f 2Z[x1;x2; : : : ;xd] and A is contained in a given subset R of a commutative
ring, then let f(A;A;: : : ;A) (with k arguments) denote the set of all terms
f(a1;a2; : : : ;ak) where the ai's belong to A. The polynomial f is called an
expander if there exists >0 such that jf(A;: : : ;A)j>jAj1+ for any nite
set A, where jBj denotes the cardinality of a nite set B. If R is nite, as
for instance, if R=Fq or f1; : : : ;Ng, we need to restrict the above de nition
by assuming that jRj"<jAj<jRj1", for some ">0. A more restrictive no-
tion is of a covering polynomial which arises from the following question: is
there a non trivial minimal size such that if A attains it, then f(A;A;: : : ;A)
entirely covers R? |
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