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Semi-Regular Sequences over F2

Molina Aristizabal, Sergio D
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1445342810

Abstract Details

2015, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
The concept of semi-regular sequences was introduced in order to assess the complexity of Gröumlbner basis algorithms such as F4 for the solution of polynomial equations. Despite the experimental evidence that semi-regular sequences are common, it was unknown whether there existed semi-regular sequences for all n, except in extremely trivial situations. In the present work I prove some results on the existence and non-existence of semi-regular sequences. It was observed by J. Schlather and T. Hodges that if an element of degree d in Β(n)-variables is semi-regular, then we must have n≤3d. In this thesis, I establish precisely when the elementary symmetric polynomial of degree d is semi-regular. In particular, when d=2t and n=3d, the elementary symmetric polynomial of degree d is semi-regular establishing that the bound given by J. Schlather and T. Hodges is sharp for infinitely many n. For the general case of existence of semi-regular sequences Bardet, Faug&egravere and Salvy conjecture that the proportion π(n, m, d1, . . . , dm) of semi-regular sequences over F2 in the set Ε(n, m, d1, . . . , dm) of algebraic systems of m equations of degrees d1, . . . , dm in n-variables tends to 1 as n tends to infinity. In this work, I show that for a fixed choice of (m, d1, . . . , dm), we have that limn→∞ π(n, m, d1, . . . , dm ) — 0 showing that the conjecture is false in this case.
Timothy Hodges, Ph.D. (Committee Chair)
Donald French, Ph.D. (Committee Member)
Tara Smith, Ph.D. (Committee Member)
105 p.
Mathematics
Abstract Algebra; Semi-Regular Sequences; Symmetric Polynomials; Cryptography; Regular Sequences; Systems of polynomial equations

Recommended Citations

Citations

  • Molina Aristizabal, S. D. (2015). Semi-Regular Sequences over F2 [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1445342810

    APA Style (7th edition)

  • Molina Aristizabal, Sergio. Semi-Regular Sequences over F2. 2015. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1445342810.

    MLA Style (8th edition)

  • Molina Aristizabal, Sergio. "Semi-Regular Sequences over F2." Doctoral dissertation, University of Cincinnati, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1445342810

    Chicago Manual of Style (17th edition)

ucin1445342810
339
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This open access ETD is published by University of Cincinnati and OhioLINK.