Casas-Alvero conjecture explained

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 wherep ≠ 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

0985.14012 . Casas-Alvero . Eduardo . Higher order polar germs . J. Algebra . 240 . 1 . 326–337 . 2001 . 0021-8693 . 10.1006/jabr.2000.8727. free .
Book: 1108.65046 . Diaz-Toca . Gema M. . Gonzalez-Vega . Laureano . On analyzing a conjecture about univariate polynomials and their roots by using Maple . Kotsireas . Ilias . Maple conference 2006. Proceedings of the conference, Waterloo, Ontario, Canada, July 23–26, 2006 . Waterloo . . 81–98 . 2006 . 1-897310-13-7 .
1127.12002 . Graf von Bothmer . Hans-Christian . Labs . Oliver . Schicho . Josef . van de Woestijne . Christiaan . The Casas-Alvero conjecture for infinitely many degrees . J. Algebra . 316 . 1 . 224–230 . 2007 . 10.1016/j.jalgebra.2007.06.017. math/0605090 . 11623853 .
1292.12001 . Draisma . Jan . de Jong . Johan P. . On the Casas-Alvero conjecture . Eur. Math. Soc. Newsl. . 80 . 29–33 . 2011 . 1027-488X . dead . https://web.archive.org/web/20160304090949/http://www.ems-ph.org/journals/newsletter/pdf/2011-06-80.pdf . 2016-03-04 .
1208.5404 . Constraints on counterexamples to the Casas-Alvero conjecture, and a verification in degree 12 . math.AG . 2012 . Wouter . Castryck . Robert . Laterveer . Myriam . Ounaïes .