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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. 3

%I

%S 0,1,1,2,4,2,4,8,4,2,24,2

%N 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 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.

%H 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 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 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 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 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 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 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).

%t See the paper by Lebl and Lichtblau.

%Y Cf. A143106, A143108, A143109.

%K nonn,more

%O 1,4

%A _Jiri Lebl_, Jul 25 2008

%E One more term (24), added addendum to and corrected title of paper - _Jiri Lebl_, Feb 08 2013

%E Added another term (2) that was computed in the newer version of the addendum. Edited by _Jiri Lebl_, May 02 2014

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