cp's OEIS Frontend

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.

%I A143107 #42 Aug 12 2025 19:54:21
%S A143107 0,1,1,2,4,2,4,8,4,2,24,2
%N 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.
%C A143107 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.
%C A143107 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
%H A143107 C. Améndola, V. Nguyen and J. Oldekop, <a href="https://arxiv.org/abs/2507.18686">One-dimensional discrete models of maximum likelihood degree one</a>, arXiv:2507.18686 [math.ST], 2025.
%H A143107 A. Bik and O. Marigliano, <a href="https://doi.org/10.1016/j.aam.2025.102928">Classifying one-dimensional discrete models with maximum likelihood degree one</a>, Adv. Appl. Math., 170 (2025), 102928.
%H A143107 J. P. D'Angelo and J. Lebl, <a href="http://dx.doi.org/10.1142/S0129167X09005248">Complexity results for CR mappings between spheres</a>, Internat. J. Math. 20 (2009), no. 2, 149-166.
%H A143107 J. P. D'Angelo and J. Lebl, <a href="http://arXiv.org/abs/0708.3232">Complexity results for CR mappings between spheres</a>, arXiv:0708.3232 [math.CV], 2008.
%H A143107 J. P. D'Angelo, Simon Kos and Emily Riehl, <a href="http://dx.doi.org/10.1007/BF02921879">A sharp bound for the degree of proper monomial mappings between balls</a>, J. Geom. Anal., 13(4):581-593, 2003.
%H A143107 J. Lebl, <a href="http://arxiv.org/abs/1302.1441">Addendum to Uniqueness of certain polynomials constant on a line</a> arxiv 1302.1441 [math.AC], 2013.
%H A143107 J. Lebl and D. Lichtblau, <a href="http://arxiv.org/abs/0808.0284">Uniqueness of certain polynomials constant on a hyperplane</a>, arXiv:0808.0284 [math.CV], 2008-2010.
%H A143107 J. Lebl and D. Lichtblau, <a href="http://dx.doi.org/10.1016/j.laa.2010.04.020">Uniqueness of certain polynomials constant on a hyperplane</a>, Linear Algebra Appl., 433 (2010), no. 4, 824-837.
%e A143107 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).
%e A143107 This corresponds to the discrete model on 3 states parametrized by t-> (t^3, 3t(1-t), (1-t)^3), 0<=t<=1.
%t A143107 See the paper by Lebl and Lichtblau.
%Y A143107 Cf. A143106, A143108, A143109.
%K A143107 nonn,more
%O A143107 1,4
%A A143107 _Jiri Lebl_, Jul 25 2008
%E A143107 One more term (24), added addendum to and corrected title of paper - _Jiri Lebl_, Feb 08 2013
%E A143107 Added another term (2) that was computed in the newer version of the addendum. Edited by _Jiri Lebl_, May 02 2014