A143107 Let H(2,d) be the space of polynomials p(x,y) of two variables with nonnegative coefficients such that p(x,y) = 1 whenever x + y = 1; a(n) is the number of different polynomials in H(2,d) with exactly n distinct monomials and of maximum degree, i.e., of degree 2n-3.

0, 1, 1, 2, 4, 2, 4, 8, 4, 2, 24, 2

Offset: 1

Author

Jiri Lebl, Jul 25 2008

Keywords

• nonn
• more

Comments

It is unknown if this sequence is bounded. For all n >= 4, a(n) is at least two. It is unknown if it is 2 for infinitely many n. It is unknown if it is always even for all n >= 2. Note that 2n-3 appears in A143106 if and only if a(n) is 1 or 2.

Also the number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n states and of degree 2n-3. - Carlos Améndola, Aug 05 2025

Examples

			a(3) = 1 as x^3 + 3xy + y^3 is the unique polynomial in H(2,d) with 3 terms and of maximum degree (in this case 3).
This corresponds to the discrete model on 3 states parametrized by t-> (t^3, 3t(1-t), (1-t)^3), 0<=t<=1.
		

Links

• C. Améndola, V. Nguyen and J. Oldekop, One-dimensional discrete models of maximum likelihood degree one, arXiv:2507.18686 [math.ST], 2025.
• A. Bik and O. Marigliano, Classifying one-dimensional discrete models with maximum likelihood degree one, Adv. Appl. Math., 170 (2025), 102928.
• 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 line arxiv 1302.1441 [math.AC], 2013.
• J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, arXiv:0808.0284 [math.CV], 2008-2010.
• J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, Linear Algebra Appl., 433 (2010), no. 4, 824-837.

Crossrefs

Cf. A143106, A143108, A143109.

Programs

Mathematica

See the paper by Lebl and Lichtblau.

Extensions

One more term (24), added addendum to and corrected title of paper - Jiri Lebl, Feb 08 2013

Added another term (2) that was computed in the newer version of the addendum. Edited by Jiri Lebl, May 02 2014