Casas-Alvero conjecture

Dr. Casas-Alvero talking about the conjecture in a conference in the University of Barcelona on March 16, 2016.

In mathematics, the Casas-Alvero conjecture is an open problem about polynomials which have factors in common with their derivatives, proposed by Eduardo Casas-Alvero in 2001.

Formal statement

Let f be a polynomial of degree d defined over a field K of characteristic zero. If f has a factor in common with each of its derivatives f⁽ⁱ⁾, i = 1, ..., d − 1, then the conjecture predicts that f must be a power of a linear polynomial.

Analogue in non-zero characteristic

The conjecture is false over a field of characteristic p: any inseparable polynomial f(X^p) without constant term satisfies the condition since all derivatives are zero. Another counterexample (which is separable) is X^p+1 − X^p.

Special cases

The conjecture is known to hold in characteristic zero for degrees of the form p^k or 2p^k where p is prime and k is a positive integer. Similarly, it is known for degrees of the form 3p^k where p ≠ 2, for degrees of the form 4p^k where p ≠ 3, 5, 7, and for degrees of the form 5p^k where p ≠ 2, 3, 7, 11, 131, 193, 599, 3541, 8009. Similar results are available for degrees of the form 6p^k and 7p^k. It has recently been established for d = 12, making d = 20 the smallest open degree.

References

Casas-Alvero, Eduardo (2001). "Higher order polar germs". J. Algebra. 240 (1): 326–337. doi:10.1006/jabr.2000.8727. ISSN 0021-8693. Zbl 0985.14012.
Diaz-Toca, Gema M.; Gonzalez-Vega, Laureano (2006). "On analyzing a conjecture about univariate polynomials and their roots by using Maple". In Kotsireas, Ilias (ed.). Maple conference 2006. Proceedings of the conference, Waterloo, Ontario, Canada, July 23–26, 2006. Waterloo: Maplesoft. pp. 81–98. ISBN 1-897310-13-7. Zbl 1108.65046.
Graf von Bothmer, Hans-Christian; Labs, Oliver; Schicho, Josef; van de Woestijne, Christiaan (2007). "The Casas-Alvero conjecture for infinitely many degrees". J. Algebra. 316 (1): 224–230. arXiv:math/0605090. doi:10.1016/j.jalgebra.2007.06.017. S2CID 11623853. Zbl 1127.12002.
Draisma, Jan; de Jong, Johan P. (2011). "On the Casas-Alvero conjecture" (PDF). Eur. Math. Soc. Newsl. 80: 29–33. ISSN 1027-488X. Zbl 1292.12001. Archived from the original (PDF) on 2016-03-04.
Castryck, Wouter; Laterveer, Robert; Ounaïes, Myriam (2012). "Constraints on counterexamples to the Casas-Alvero conjecture, and a verification in degree 12". arXiv:1208.5404 [math.AG].