On solving norm equations in global function fields. (English) Zbl 1233.11130
Let \(E/F\) be a finite extension of fields of degree \(d\), where \(F=\mathbb{Q}\) or \(\mathbb{F}_q(t)\) with \(q=p^l\), \(p\) a prime. Let \(\gamma_1, \dots , \gamma_m\) be \(m\) \(F\)-linearly independent elements of \(E\) with \(2\leq m \leq d\) and the norm equation
\[
N(x)=c,
\]
where \(x=x_1 \gamma_1 + \dots + x_m \gamma_m\), \(c, x_i\in o_F\).
The computation of solutions of this equation is well-known when \(m=2\) and \(m=d\), particularly for algebraic number fields. For example, see C. Fieker, A. Jurk and M. Pohst [Math. Comput. 66, No. 217, 399–410 (1997; Zbl 0854.11076)], D. Simon [Math. Comput. 71, No. 239, 1287–1305 (2002; Zbl 0990.11014)], Y. Bilu and G. Hanrot [J. Number Theory 60, No. 2, 373–392 (1996; Zbl 0867.11017)], and I. Gaál and M. Pohst [Exp. Math. 15, No. 1, 1–6 (2006; Zbl 1142.11019)].
In this paper, the authors extend their work done for \(m=3\) to arbitrary \(m\), see [J. Number Theory 130, No. 3, 493–506 (2010; Zbl 1243.11120)]. They develop the theory for computing all solutions of the equation. Moreover, they give a example with \(E=k(t)(\alpha)\), where \(k=\mathbb{F}_5\) and \(\alpha\) is a zero of \(p(z) = z^5-z-t\).
The computation of solutions of this equation is well-known when \(m=2\) and \(m=d\), particularly for algebraic number fields. For example, see C. Fieker, A. Jurk and M. Pohst [Math. Comput. 66, No. 217, 399–410 (1997; Zbl 0854.11076)], D. Simon [Math. Comput. 71, No. 239, 1287–1305 (2002; Zbl 0990.11014)], Y. Bilu and G. Hanrot [J. Number Theory 60, No. 2, 373–392 (1996; Zbl 0867.11017)], and I. Gaál and M. Pohst [Exp. Math. 15, No. 1, 1–6 (2006; Zbl 1142.11019)].
In this paper, the authors extend their work done for \(m=3\) to arbitrary \(m\), see [J. Number Theory 130, No. 3, 493–506 (2010; Zbl 1243.11120)]. They develop the theory for computing all solutions of the equation. Moreover, they give a example with \(E=k(t)(\alpha)\), where \(k=\mathbb{F}_5\) and \(\alpha\) is a zero of \(p(z) = z^5-z-t\).
Reviewer: Alain S. Togbe (Westville)
MSC:
| 11Y50 | Computer solution of Diophantine equations |
| 11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |
| 11D57 | Multiplicative and norm form equations |
| 94A60 | Cryptography |
References:
| [1] | DOI: 10.1006/jnth.1996.0129 · Zbl 0867.11017 · doi:10.1006/jnth.1996.0129 |
| [2] | DOI: 10.1006/jsco.1996.0126 · Zbl 0886.11070 · doi:10.1006/jsco.1996.0126 |
| [3] | DOI: 10.1090/S0025-5718-97-00761-8 · Zbl 0854.11076 · doi:10.1090/S0025-5718-97-00761-8 |
| [4] | DOI: 10.1016/j.jnt.2005.10.009 · Zbl 1157.11011 · doi:10.1016/j.jnt.2005.10.009 |
| [5] | Exp. Math. 15 pp 1– (2006) |
| [6] | DOI: 10.1016/j.jsc.2005.03.003 · Zbl 1156.11347 · doi:10.1016/j.jsc.2005.03.003 |
| [7] | DOI: 10.1090/S0025-5718-02-01309-1 · Zbl 0990.11014 · doi:10.1090/S0025-5718-02-01309-1 |
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