# 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 & 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. [2] G. P. Barker, Theory of cones,, Linear Algebra and its Applications, 39 (1981), 263. 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. doi: 10.1007/BF02574037. [4] J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry,, Springer-Verlag, (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. [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). [8] J. S. Geronimo and M.-J. Lai, Factorization of multivariate positive Laurent polynomials,, Journal of Approximation Theory, 139 (2006), 327. doi: 10.1016/j.jat.2005.09.010. [9] O. Güler, Barrier function in interior point methods,, Mathematics of Operations Research, 21 (1996), 860. 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. 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. 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. 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. 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. [16] G. Pataki, Cone-LP's and semidefinite programs: Geometry and a simplex-type method,, in Integer Programming and Combinatorial Optimization, 1084 (1996), 162. 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. doi: 10.1287/moor.23.2.339. [18] G. Pataki, The Geometry of Semidefinite Programming,, in Handbook of Semidefinite Programming: Theory, (2000). doi: 10.1007/978-1-4615-4381-7. [19] A. Prestel and C. N. Delzell, Positive Polynomials,, Springer Monographs in Mathematics, (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. [2] G. P. Barker, Theory of cones,, Linear Algebra and its Applications, 39 (1981), 263. 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. doi: 10.1007/BF02574037. [4] J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry,, Springer-Verlag, (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. [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). [8] J. S. Geronimo and M.-J. Lai, Factorization of multivariate positive Laurent polynomials,, Journal of Approximation Theory, 139 (2006), 327. doi: 10.1016/j.jat.2005.09.010. [9] O. Güler, Barrier function in interior point methods,, Mathematics of Operations Research, 21 (1996), 860. 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. 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. 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. 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. 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. [16] G. Pataki, Cone-LP's and semidefinite programs: Geometry and a simplex-type method,, in Integer Programming and Combinatorial Optimization, 1084 (1996), 162. 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. doi: 10.1287/moor.23.2.339. [18] G. Pataki, The Geometry of Semidefinite Programming,, in Handbook of Semidefinite Programming: Theory, (2000). doi: 10.1007/978-1-4615-4381-7. [19] A. Prestel and C. N. Delzell, Positive Polynomials,, Springer Monographs in Mathematics, (2001). doi: 10.1007/978-3-662-04648-7. [20] R. T. Rockafellar, Convex Analysis,, Princeton University Press, (1970).
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