Advertisement

Combinatorica

, Volume 14, Issue 2, pp 149–157 | Cite as

On the number of nowhere zero points in linear mappings

  • R. D. Baker
  • J. Bonin
  • F. Lazebnik
  • E. Shustin
Article

Abstract

LetA be a nonsingularn byn matrix over the finite fieldGF q ,k=⌊n/2⌋,q=p a ,a≥1, wherep is prime. LetP(A,q) denote the number of vectorsx in (GF q ) n such that bothx andAx have no zero component. We prove that forn≥2, and\(q > 2\left( {\begin{array}{*{20}c} {2n} \\ 3 \\ \end{array} } \right)\),P(A,q)≥[(q−1)(q−3)] k (q−2) n−2k and describe all matricesA for which the equality holds. We also prove that the result conjectured in [1], namely thatP(A,q)≥1, is true for allq≥n+2≥3 orq≥n+1≥4.

AMS subject classification code (1991)

06 C 10 15 A 06 11 T 99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. Alon, andM. Tarsi: A nowhere zero point in linear mappings,Combinatorica 9(4) (1989), 393–395.Google Scholar
  2. [2]
    M. Fujiwara: Über die obere Schranke des absoluten Betrages der Wurzeln einer algebraischen Gleichung,Tôhoku Math. J. 10 (1916), 167–171.Google Scholar
  3. [3]
    G.-C. Rota: On the Foundations of Combinatorial Theory I: Theory of Möbius Functions,Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, 340–368, 1964.Google Scholar
  4. [4]
    H. S. Wilf:Mathematics for the Physical Sciences (Wiley, New York, 1978).Google Scholar

Copyright information

© Akadémiai Kiadó 1994

Authors and Affiliations

  • R. D. Baker
    • 1
  • J. Bonin
    • 2
  • F. Lazebnik
    • 1
  • E. Shustin
    • 3
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Department of MathematicsThe George Washington UniversityWashington, DCUSA
  3. 3.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations