American Institute of Mathematical Sciences

2016, 6(2): 91-102. doi: 10.3934/naco.2016001

On bounds of the Pythagoras number of the sum of square magnitudes of Laurent polynomials

 1 Department of Mathematics, Quy Nhon University, Vietnam 2 Department of Computer Science, KU Leuven, Belgium

Received  October 2014 Revised  March 2016 Published  June 2016

This paper presents a lower and upper bound of the Pythagoras number of sum of square magnitudes of Laurent polynomials (sosm-polynomials). To prove these bounds, properties of the corresponding system of quadratic polynomial equations are used. Applying this method, a new proof for the best (known until now) upper bound of the Pythagoras number of real polynomials is also presented.
Citation: Thanh Hieu Le, Marc Van Barel. On bounds of the Pythagoras number of the sum of square magnitudes of Laurent polynomials. Numerical Algebra, Control and Optimization, 2016, 6 (2) : 91-102. doi: 10.3934/naco.2016001
References:
 [1] G. P. Barker, The lattice of faces of a finite dimensional cone, Linear Algebra and its Applications, 7 (1973), 71-82. [2] G. P. Barker, Theory of cones, Linear Algebra and its Applications, 39 (1981), 263-291. doi: 10.1016/0024-3795(81)90310-4. [3] A. I. Barvinok, Problems of distance geometry and convex properties of quadratic maps, Discrete & Computational Geometry, 13 (1995), 189-202. doi: 10.1007/BF02574037. [4] J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Springer-Verlag, Berlin, Heidelberg, 1998. doi: 10.1007/978-3-662-03718-8. [5] S. P. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. doi: 10.1017/CBO9780511804441. [6] M.-D. Choi, T. Y. Lam and B. Reznick, Sums of squares of real polynomials, in Proceedings of Symposia in Pure Mathematics, 58 (1995), 103-126. [7] Y. V. Genin, Y. Hachez, Y. Nesterov and P. Van Dooren, Convex optimization over positive polynomials and filter design, in Proceedings UKACC Int. Conf. Control 2000, 2000, Paper SS41. [8] J. S. Geronimo and M.-J. Lai, Factorization of multivariate positive Laurent polynomials, Journal of Approximation Theory, 139 (2006), 327-345. doi: 10.1016/j.jat.2005.09.010. [9] O. Güler, Barrier function in interior point methods, Mathematics of Operations Research, 21 (1996), 860-885. doi: 10.1287/moor.21.4.860. [10] R. D. Hill and S. R. Waters, On the cone of positive semidefinite matrices, Linear Algebra and its Applications, 90 (1987), 81-88. doi: 10.1016/0024-3795(87)90307-7. [11] W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, 1948. [12] J. B. Lasserre, A sum of squares approximation of nonnegative polynomials, SIAM Review, 49 (2007), 651-669. doi: 10.1137/070693709. [13] T. H. Le, L. Sorber and M. Van Barel, The Pythagoras number of real sum of squares polynomials and sum of square magnitudes of polynomials, Calcolo, 50 (2013), 283-303. doi: 10.1007/s10092-012-0068-y. [14] T. H. Le and M. Van Barel, A convex optimization method to solve a filter design problem, Journal of Computational and Applied Mathematics, 255 (2014), 183-192. doi: 10.1016/j.cam.2013.04.044. [15] G. Marsaglia and G. P. H. Styan, When does rank(A+B) = rank(A) + rank(B)?, Canadian Mathematical Bulletin, 15 (1972), 451-452. [16] G. Pataki, Cone-LP's and semidefinite programs: Geometry and a simplex-type method, in Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, 1084 (1996), 162-174. doi: 10.1007/3-540-61310-2_13. [17] G. Pataki, On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues, Mathematics of Operations Research, 23 (1998), 339-358. doi: 10.1287/moor.23.2.339. [18] G. Pataki, The Geometry of Semidefinite Programming, in Handbook of Semidefinite Programming: Theory, Algorithms, and Applications (eds. H. Wolkowicz, R. Saigal and L. Vandenberghe), International series in operations research and management science, Kluwer Academic Publishers, 2000. doi: 10.1007/978-1-4615-4381-7. [19] A. Prestel and C. N. Delzell, Positive Polynomials, Springer Monographs in Mathematics, Springer, 2001. doi: 10.1007/978-3-662-04648-7. [20] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.

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References:
 [1] G. P. Barker, The lattice of faces of a finite dimensional cone, Linear Algebra and its Applications, 7 (1973), 71-82. [2] G. P. Barker, Theory of cones, Linear Algebra and its Applications, 39 (1981), 263-291. doi: 10.1016/0024-3795(81)90310-4. [3] A. I. Barvinok, Problems of distance geometry and convex properties of quadratic maps, Discrete & Computational Geometry, 13 (1995), 189-202. doi: 10.1007/BF02574037. [4] J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Springer-Verlag, Berlin, Heidelberg, 1998. doi: 10.1007/978-3-662-03718-8. [5] S. P. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. doi: 10.1017/CBO9780511804441. [6] M.-D. Choi, T. Y. Lam and B. Reznick, Sums of squares of real polynomials, in Proceedings of Symposia in Pure Mathematics, 58 (1995), 103-126. [7] Y. V. Genin, Y. Hachez, Y. Nesterov and P. Van Dooren, Convex optimization over positive polynomials and filter design, in Proceedings UKACC Int. Conf. Control 2000, 2000, Paper SS41. [8] J. S. Geronimo and M.-J. Lai, Factorization of multivariate positive Laurent polynomials, Journal of Approximation Theory, 139 (2006), 327-345. doi: 10.1016/j.jat.2005.09.010. [9] O. Güler, Barrier function in interior point methods, Mathematics of Operations Research, 21 (1996), 860-885. doi: 10.1287/moor.21.4.860. [10] R. D. Hill and S. R. Waters, On the cone of positive semidefinite matrices, Linear Algebra and its Applications, 90 (1987), 81-88. doi: 10.1016/0024-3795(87)90307-7. [11] W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, 1948. [12] J. B. Lasserre, A sum of squares approximation of nonnegative polynomials, SIAM Review, 49 (2007), 651-669. doi: 10.1137/070693709. [13] T. H. Le, L. Sorber and M. Van Barel, The Pythagoras number of real sum of squares polynomials and sum of square magnitudes of polynomials, Calcolo, 50 (2013), 283-303. doi: 10.1007/s10092-012-0068-y. [14] T. H. Le and M. Van Barel, A convex optimization method to solve a filter design problem, Journal of Computational and Applied Mathematics, 255 (2014), 183-192. doi: 10.1016/j.cam.2013.04.044. [15] G. Marsaglia and G. P. H. Styan, When does rank(A+B) = rank(A) + rank(B)?, Canadian Mathematical Bulletin, 15 (1972), 451-452. [16] G. Pataki, Cone-LP's and semidefinite programs: Geometry and a simplex-type method, in Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, 1084 (1996), 162-174. doi: 10.1007/3-540-61310-2_13. [17] G. Pataki, On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues, Mathematics of Operations Research, 23 (1998), 339-358. doi: 10.1287/moor.23.2.339. [18] G. Pataki, The Geometry of Semidefinite Programming, in Handbook of Semidefinite Programming: Theory, Algorithms, and Applications (eds. H. Wolkowicz, R. Saigal and L. Vandenberghe), International series in operations research and management science, Kluwer Academic Publishers, 2000. doi: 10.1007/978-1-4615-4381-7. [19] A. Prestel and C. N. Delzell, Positive Polynomials, Springer Monographs in Mathematics, Springer, 2001. doi: 10.1007/978-3-662-04648-7. [20] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.
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