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Title: Math/Number Theory - Somos Polynomials Related to Somos sequences and elliptic theta functions.
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Somos Polynomials

Somos Polynomials

Define a sequence with:S(1) = x , S(2) = S(3) = 1 , S(4) = y ,S(n) = x*y*z*(S(n-1)*S(n-3)+S(n-2)^2)/S(n-4) if n > 4 .The first observation is that this definition does producepolynomials with positive integer coefficients in the threeindeterminates x , y , z. The proof is by induction.Another observation is that S(n) can be easily extended ton < 1 so that it still satisfies the non-linear equation.When S(1)=S(2)=S(3)=S(4)=1, then S(n) becomes the Somos-4sequence of integers.For more information about these sequences, see article"SomosSequence" in MathWorld.So far, I have not found a way to extend this to other Somossequences such as the Somos 6 sequence.Finally, this polynomial sequence is universalin the sense that it can be used to express any solutionof the general Theta-4 recursion:a(n) = (p1*a(n-1)*a(n-3)+p2*a(n-2)^2)/a(n-4) .A brief table of the polynomials is:S(1) = xS(2) = 1S(3) = 1S(4) = yS(5) = y^2*z + y*zS(6) = x*y^3*z^2 + x*y^3*z + x*y^2*z^2S(7) = x^2*y^5*z^3 + x^2*y^5*z^2 + x^2*y^4*z^3 + x*y^5*z^3 + 2*x*y^4*z^3 + x*y^3*z^3S(8) = x^3*y^7*z^5 + x^3*y^7*z^4 + 3*x^3*y^6*z^5 + 3*x^3*y^6*z^4 + x^3*y^6*z^3 + 3*x^3*y^5*z^5 + 2*x^3*y^5*z^4 + x^3*y^4*z^5 + x^2*y^7*z^5 + 3*x^2*y^6*z^5 + 3*x^2*y^5*z^5 + x^2*y^4*z^5If the maximum exponent of x in all terms of S(n) is taken, the sequence isn 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23e(n) 1 0 0 0 0 1 2 3 5 7 9 12 15 18 22 26 30 35 40 45 51 57 63which is given by the following simple recursion:e(n) = 1 + e(n-1) + e(n-3) - e(n-4) .The maximum exponent sequence of y is e(n+2), and of z is e(n+1). The sum ofthe maximum exponents of x, y, and z is (n-2)*(n-3)/2, not coincidently atriangular number.Another shifted version of the e(n) sequence is listed as A001840 in Sloane's Encyclopedia of Integer Sequences. It has a rational generating funcion.Based on earlier work, around December 1992 I came up with thissequence of polynomials with unusual properties. An explanation of the wayin which this sequence was discovered is a long story, but itstarted with a study of elliptic theta functions. More about this real soon.Back to my math pageBack to my home pageLast Updated Wed Aug 6 04:36 EDT 2003Michael Somos <somos@grail.cba.csuohio.edu>WWW URL: "http://grail.cba.csuohio.edu/~somos/"
 

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