TY - JOUR

T1 - On the number of nowhere zero points in linear mappings

AU - Baker, R. D.

AU - Bonin, J.

AU - Lazebnik, F.

AU - Shustin, E.

PY - 1994/6

Y1 - 1994/6

N2 - Let A be a nonsingular n by n matrix over the finite field GFq, k=⌊n/2⌋, q=pa, a≥1, where p is prime. Let P(A,q) denote the number of vectors x in (GFq)n such that both x and Ax have no zero component. We prove that for n≥2, and {Mathematical expression}, P(A,q)≥[(q-1)(q-3)]k(q-2)n-2k and describe all matrices A for which the equality holds. We also prove that the result conjectured in [1], namely that P(A,q)≥1, is true for all q≥n+2≥3 or q≥n+1≥4.

AB - Let A be a nonsingular n by n matrix over the finite field GFq, k=⌊n/2⌋, q=pa, a≥1, where p is prime. Let P(A,q) denote the number of vectors x in (GFq)n such that both x and Ax have no zero component. We prove that for n≥2, and {Mathematical expression}, P(A,q)≥[(q-1)(q-3)]k(q-2)n-2k and describe all matrices A for which the equality holds. We also prove that the result conjectured in [1], namely that P(A,q)≥1, is true for all q≥n+2≥3 or q≥n+1≥4.

KW - AMS subject classification code (1991): 06C10, 15A06, 11T99

UR - http://www.scopus.com/inward/record.url?scp=0008990495&partnerID=8YFLogxK

U2 - 10.1007/BF01215347

DO - 10.1007/BF01215347

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AN - SCOPUS:0008990495

SN - 0209-9683

VL - 14

SP - 149

EP - 157

JO - Combinatorica

JF - Combinatorica

IS - 2

ER -