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

1,4

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

Links

Table of n, a(n) for n=1..12.

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.

Example

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.

Mathematica

See the paper by Lebl and Lichtblau.

Crossrefs

Cf. A143106, A143108, A143109.

Sequence in context: A286536 A318768 A166242 * A051638 A286580 A286598

Adjacent sequences: A143104 A143105 A143106 * A143108 A143109 A143110

Keyword

nonn,more

Author

Jiri Lebl, Jul 25 2008

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

Status

approved