Jiri Lebl - cp's OEIS frontend

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

Jiri Lebl, Jul 25 2008

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.

			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).

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

Cf. A143105, A143107.

Mathematica
```
See the paper by Lebl-Lichtblau
```

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

A143105 Let g_0(x,y)=x, g_1(x,y)=x^3+3xy and g_{n+2}(x,y) = (x^2+2y)g_{n+1}(x+y)-y^2g_n(x,y). The entries of the sequence are those odd d for which g_d(x,y) and cx^jy^kg_m(x,y) have at least two terms in common (same coefficients) for some c > 0 and integers j,k and such that g_d(x,y) + cx^jy^k(1+y^m - g_m(x,y)) has all positive coefficients.

Original entry on oeis.org

1, 3, 5, 9, 17, 21, 33, 41, 45, 53, 69, 77, 81, 93, 105, 113, 117, 125, 129, 141, 149, 153, 161, 165, 177, 185, 201, 213, 221, 225, 249, 261, 269, 273, 285, 297, 305, 309, 333, 341, 345, 357, 365, 369, 381, 405, 413, 417, 429, 437, 441, 453, 465, 473, 489, 501

Offset: 0

Jiri Lebl (jlebl(AT)math.uiuc.edu), Jul 25 2008

nonn

Note that g_k(x,y) always has positive coefficients. The sequence are degrees for which a certain construction (see paper by D'Angelo-Lebl) of proper monomial holomorphic mappings of balls does not give new noninvariant monomial mappings.

It is unknown if this sequence is infinite (conjectured to be so). Furthermore A143106 is definitely a subsequence of this sequence, but it is unknown if the two are in fact equal.

			For example when d=7, we get the following new polynomials x^7 + 7/2 xy + 7/2 x^5y 7/2 xy^5 and x^7 + 7 x^3y + 7 xy^3 + 7 x^3y^3. Hence 7 is not in the sequence.

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) (2003) 581-593.

J. P. D'Angelo and J. Lebl. Complexity results for CR mappings between spheres, arXiv:0708.3232, Internat. J. Math. 20 (2) (2009) 149
J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, arXiv:0808.0284 (2008).

Cf. A143106.

Mathematica
```
See the paper by Lebl-Lichtblau
```

A143108 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 minus 1, i.e., of degree 2n-4.

Original entry on oeis.org

0, 0, 3, 4, 10, 24, 32, 56

Offset: 1

Jiri Lebl (jlebl(AT)math.uiuc.edu), Jul 25 2008

a(n) is also the number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n states and of degree 2n-4. - Carlos Améndola, Aug 05 2025

Carlos Améndola, Viet Duc Nguyen, and Janike Oldekop, One-dimensional Discrete Models of Maximum Likelihood Degree One, arXiv:2507.18686 [math.ST], 2025. See p. 24.
John 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.
John P. D'Angelo and Jiří Lebl, Complexity results for CR mappings between spheres, arXiv:0708.3232 [math.CV], 2008.
John P. D'Angelo and Jiří Lebl, Complexity results for CR mappings between spheres, Internat. J. Math. 20 (2009), no. 2, 149-166.
Jiří Lebl and Daniel Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, arXiv:0808.0284 [math.CV], 2008-2010.
Jiří Lebl and Daniel Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, Linear Algebra Appl., 433 (2010), no. 4, 824-837

Cf. A143107, A143109.

Mathematica
```
See the paper by Lebl-Lichtblau.
```

Possibly can be computed from A143107 except for the third term, but this is not proved. Let b_n be elements of A143107, then a_n = 2 ( b_2 b_{n-1} + b_3 b_{n-2} + ... + b_{n-1} b_2 ).

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.

Original entry on oeis.org

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

Offset: 1

Jiri Lebl, Jul 25 2008

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

			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.

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.

Cf. A143106, A143108, A143109.

Mathematica
```
See the paper by Lebl and Lichtblau.
```

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

A143109 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 minus two, i.e., of degree 2n-5.

Original entry on oeis.org

0, 0, 0, 11, 38, 88, 198

Offset: 1

Jiri Lebl (jlebl(AT)math.uiuc.edu), Jul 25 2008

It is unknown but conjectured that this is a sequence of finite numbers. Note that if we went one degree lower and look at polynomials of degree 2n-6, then there are infinitely many if any exist in H(2,d).

Likely an erroneous version of A387029. - Sean A. Irvine, Aug 13 2025

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. P. D'Angelo and J. Lebl, Complexity results for CR mappings between spheres, arXiv:0708.3232 [math.CV], 2008.
J. P. D'Angelo and J. Lebl, Complexity results for CR mappings between spheres, Internat. J. Math. 20 (2009), no. 2, 149-166.
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

Cf. A143107, A143108, A387029.

Mathematica
```
(* See the paper by Lebl-Lichtblau. *)
```

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A143108 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 minus 1, i.e., of degree 2n-4.

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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.

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A143109 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 minus two, i.e., of degree 2n-5.

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