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Polynomial multiplicities over finite fields and intersection sets. (English) Zbl 0754.05024

Several years ago a major breakthrough in the study of intersection sets (1-intersection sets) in affine spaces over finite fields was obtained by R. E. Jamison [J. Comb. Theory, Ser. A 22, 253-266 (1977; Zbl 0354.12019)]. Subsequently, an alternative treatment of some of Jamison’s results was given by A. E. Brouwer and A. Schrijver [J. Comb. Theory, Ser. A 24, 251-253 (1978; Zbl 0373.05020)].
We generalize the results in [(*) R. E. Jamison, J. Comb. Theory, Ser. A 22, 253-266 (1977; Zbl 0353.12019)], [(**) A. E. Brouwer, A. Schrijver, J. Comb. Theory, Ser. A 24, 251-253 (1978; Zbl 0372.05020)] in several different directions, although our work is connected with some very general questions concerning certain polynomials in several variables over Galois fields. In particular, we obtain lower bounds on \(t\)-intersection sets and on \(t\)-coverings by subspaces. In the special case when the subspaces are in fact hyperplanes we offer two different proofs, one based on the method in [(**)] (Sections 1,2), the other on the ideas in [(*)] (Sections 3,4). Moreover, the work in Section 3 also provides a significant simplification of the proof in [(*)].
Our results in Section 1 connect the \(t\)-intersection question with a general lower bound, obtained by elementary methods, on the degree of a polynomial in several variables over \(GF(q)\) having certain multiplicity properties (Theorem 1.4). We also point out what appears to be an interesting and surprising connection between the attainment of this lower bound and the existence of certain well-studied configurations known as arcs in projective spaces. In Section 4 we discuss further applications to filling the points of \(AG(2,q)\) with curves of a certain degree and, also, to intersection sets in non-Desarguesian planes.

MSC:

05B25 Combinatorial aspects of finite geometries
12E10 Special polynomials in general fields
51A40 Translation planes and spreads in linear incidence geometry
12E20 Finite fields (field-theoretic aspects)
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[1] Brouwer, A. E.; Schrijver, A., The blocking number of an affine space, J. Combin. Theory Ser. A, 24, 251-253 (1978) · Zbl 0373.05020
[2] Bruen, A. A., Arcs and multiple blocking sets, (Symposia Mathematica, Vol. 28, Instituto Nazionale di Alta Matematica Fracesco Severi (1986), Academic Press: Academic Press New York/London) · Zbl 0472.51006
[3] Bruen, A. A., Blocking sets, abstract of one hour invited address at Combinatorics ’84 (1984), University of Bari
[4] Bruen, A. A.; Thas, J. A.; Blokhuis, A., On M.D.S. codes, arcs in \(PG (n, q)\) with \(q\) even, and a solution of three fundamental problems of B. Segre, Invent. Math., 92, 441-459 (1988) · Zbl 0654.94014
[5] Greenberg, M. J., (Lectures on Forms in Many Variables (1969), Benjamin: Benjamin New York) · Zbl 0185.08304
[6] Jacobson, N., (Lectures in Abstract Algebra, Vol. 1 (1951), Van Nostrand: Van Nostrand New York)
[7] Jamison, R. E., Covering finite fields with cosets of subspaces, J. Combin. Theory Ser. A, 22, 253-266 (1977) · Zbl 0354.12019
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