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Representation numbers of five sextenary quadratic forms

Ernest X. W. Xia, Olivia X. M. Yao, A. F. Y. Zhao (2015)

Colloquium Mathematicae

For nonnegative integers a, b, c and positive integer n, let N(a,b,c;n) denote the number of representations of n by the form i = 1 a ( x ² i + x i y i + y ² i ) + 2 j = 1 b ( u ² j + u j v j + v ² j ) + 4 k = 1 c ( r ² k + r k s k + s ² k ) . Explicit formulas for N(a,b,c;n) for some small values were determined by Alaca, Alaca and Williams, by Chan and Cooper, by Köklüce, and by Lomadze. We establish formulas for N(2,1,0;n), N(2,0,1;n), N(1,2,0;n), N(1,0,2;n) and N(1,1,1;n) by employing the (p, k)-parametrization of three 2-dimensional theta functions due to Alaca, Alaca and Williams.

Representations of non-negative polynomials via KKT ideals

Dang Tuan Hiep (2011)

Annales Polonici Mathematici

This paper studies the representation of a non-negative polynomial f on a non-compact semi-algebraic set K modulo its KKT (Karush-Kuhn-Tucker) ideal. Under the assumption that f satisfies the boundary Hessian conditions (BHC) at each zero of f in K, we show that f can be represented as a sum of squares (SOS) of real polynomials modulo its KKT ideal if f ≥ 0 on K.

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