“Carlitz–Wan conjecture”的意思、由来-开放百科全书

In mathematics, the Carlitz–Wan conjecture classifies the possible degrees of exceptional polynomials over a finite field F_q of q elements. A polynomial f(x) in F_q[x] of degree d is called exceptional over F_q if every irreducible factor (differing from x − y) or (f(x) − f(y))/(x − y)) over F_q becomes reducible over the algebraic closure of F_q. If q > d⁴, then f(x) is exceptional if and only if f(x) is a permutation polynomial over F_q.

The Carlitz–Wan conjecture states that there are no exceptional polynomials of degree d over F_q if gcd(d, q − 1) > 1.

In the special case that q is odd and d is even, this conjecture was proposed by Carlitz (1966) and proved by Fried, Guralnick, and Saxl (1993)^[1] The general form of the Carlitz–Wan conjecture was proposed by Daqing Wan (1993)^[2] and later proved by Hendrik Lenstra (1995)^[3]

References

1. ^{{citation|last1=Fried|first1=Michael D.|author1-link=Michael D. Fried|last2=Guralnick|first2=Robert|author2-link=Robert Guralnick|last3=Saxl|first3=Jan|doi=10.1007/BF02808112|issue=1-3|journal=Israel Journal of Mathematics|mr=1239049|pages=157–225|title=Schur covers and Carlitz's conjecture|volume=82|year=1993}}
2. ^{{citation|last=Wan|first=Daqing|authorlink=Daqing Wan|editor1-last=Mullen|editor1-first=Gary L.|editor2-last=Shiue|editor2-first=Peter Jau-Shyong|contribution=A generalization of the Carlitz conjecture|isbn=0-8247-8805-2|mr=1199817|pages=431–432|publisher=Marcel Dekker, Inc., New York|series=Lecture Notes in Pure and Applied Mathematics|title=Finite fields, Coding Theory, and Advances in Communications and Computing: Proceedings of the International Conference held at the University of Nevada, Las Vegas, Nevada, August 7–10, 1991|volume=141|year=1993}}
3. ^{{citation|last1=Cohen|first1=Stephen D.|last2=Fried|first2=Michael D.|author2-link=Michael D. Fried|doi=10.1006/ffta.1995.1027|issue=3|journal=Finite Fields and their Applications|mr=1341953|pages=372–375|title=Lenstra's proof of the Carlitz–Wan conjecture on exceptional polynomials: an elementary version|volume=1|year=1995}}