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

Author

Jiri Lebl, Jul 25 2008

Keywords

• hard
• nonn
• more

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.

Examples

			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).
		

Links

• 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

Crossrefs

Cf. A143105, A143107.

Programs

Mathematica

See the paper by Lebl-Lichtblau

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