A143106 Odd degrees for which (up to swapping of variables) there exists a unique polynomial p(x,y), such that p(x,y)=1 when x+y=1, with positive coefficients and such that the number of terms is minimal (equal to (d+3)/2). There always exists a group invariant polynomial (see any of the references), but for many degrees, other such extremal polynomials exist.

1, 3, 5, 9, 17, 21

Offset

0,2

Comments

This sequence is a subsequence of A143105. It is unknown if this is the same sequence, nor if this sequence is infinite (conjectured to be such). It is not currently computationally feasible to find out if 21 belongs in this sequence or not.

Links

Table of n, a(n) for n=0..5.

J. P. D'Angelo and J. Lebl, Complexity results for CR mappings between spheres, Internat. J. Math. 20 (2009), no. 2, 149-166.

J. P. D'Angelo and J. Lebl, Complexity results for CR mappings between spheres, arXiv:0708.3232 [math.CV], 2008.

J. P. D'Angelo, Simon Kos and Emily Riehl, A sharp bound for the degree of proper monomial mappings between balls, J. Geom. Anal., 13(4):581-593, 2003.

J. Lebl, Addendum to Uniqueness of certain polynomials constant on a hyperplane, preprint arXiv:1302:1441

J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, arXiv:0808.0284 [math.CV]

J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, Linear Algebra Appl., 433 (2010), no. 4, 824-837

Example

7 is not in the sequence as there are two noninvariant polynomials with minimal number of terms: x^7 + 7/2 xy + 7/2 x^5y + 7/2 xy^5 + y^7 and x^7 + 7 x^3y + 7 xy^3 + 7 x^3y^3 + y^7. This is beside the group invariant x^7 + 7 x^3y + 14 x^2y^3 + 7 xy^5 + y^7 (and one with x,y reversed).

Mathematica

See the paper by Lebl-Lichtblau

Crossrefs

Cf. A143105, A143107.

Sequence in context: A355611 A306973 A130114 * A364877 A217099 A276970

Adjacent sequences: A143103 A143104 A143105 * A143107 A143108 A143109

Keyword

hard,nonn,more

Author

Jiri Lebl, Jul 25 2008

Extensions

Added term 21 that was recently computed, see the recent preprint by Lebl. Added publication data for Lebl-Lichblau paper. Corrected and edited by Jiri Lebl, May 02 2014

Status

approved