Casas-Alvero conjecture

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 factor.

Analog in non-zero characteristic

The conjecture is false over a field of characteristic p: any inseparable polynomial f(X^p) satisfies the condition since all derivatives are zero. Another, separable, counterexample is X^p+1 − X^p

Special cases

The conjecture is known to hold in characteristic zero for degrees which are prime power or twice a prime power: hence for all d up to 11. It has recently been established for d = 12.

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. 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. doi:10.1016/j.jalgebra.2007.06.017. 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. 
• 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].