## A moment approach to analyze zeros of triangular polynomial sets

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- by Jean B. Lasserre PDF
- Trans. Amer. Math. Soc.
**358**(2006), 1403-1420 Request permission

## Abstract:

Let $I=\langle g_1,\ldots , g_n\rangle$ be a zero-dimensional ideal of $\mathbb {R}[x_1,\ldots ,x_n]$ such that its associated set $\mathbb {G}$ of polynomial equations $g_i(x)=0$ for all $i=1,\ldots ,n$ is in triangular form. By introducing multivariate Newton sums we provide a numerical characterization of polynomials in $\sqrt {I}$. We also provide a necessary and sufficient (numerical) condition for all the zeros of $\mathbb {G}$ to be in a given set $\mathbb {K}\subset \mathbb {C}^n$, without explicitly computing the zeros. In addition, we also provide a necessary and sufficient condition on the coefficients of the $g_i$’s for $\mathbb {G}$ to have (a) only real zeros, (b) to have only real zeros, all contained in a given semi-algebraic set $\mathbb {K}\subset \mathbb {R}^n$. In the proof technique, we use a deep result of Curto and Fialkow (2000) on the $\mathbb {K}$-moment problem, and the conditions we provide are given in terms of positive definiteness of some related moment and localizing matrices depending on the $g_i$’s via the Newton sums of $\mathbb {G}$. In addition, the number of distinct real zeros is shown to be the maximal rank of a related moment matrix.## References

- Philippe Aubry, Daniel Lazard, and Marc Moreno Maza,
*On the theories of triangular sets*, J. Symbolic Comput.**28**(1999), no. 1-2, 105–124. Polynomial elimination—algorithms and applications. MR**1709419**, DOI 10.1006/jsco.1999.0269 - Philippe Aubry and Marc Moreno Maza,
*Triangular sets for solving polynomial systems: a comparative implementation of four methods*, J. Symbolic Comput.**28**(1999), no. 1-2, 125–154. Polynomial elimination—algorithms and applications. MR**1709420**, DOI 10.1006/jsco.1999.0270 - William W. Adams and Philippe Loustaunau,
*An introduction to Gröbner bases*, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence, RI, 1994. MR**1287608**, DOI 10.1090/gsm/003 - Saugata Basu, Richard Pollack, and Marie-Françoise Roy,
*Algorithms in real algebraic geometry*, Algorithms and Computation in Mathematics, vol. 10, Springer-Verlag, Berlin, 2003. MR**1998147**, DOI 10.1007/978-3-662-05355-3 - E. Becker and T. Wörmann,
*Radical computations of zero-dimensional ideals and real root counting*, Math. Comput. Simulation**42**(1996), no. 4-6, 561–569. Symbolic computation, new trends and developments (Lille, 1993). MR**1430841**, DOI 10.1016/S0378-4754(96)00033-X - Raúl E. Curto and Lawrence A. Fialkow,
*The truncated complex $K$-moment problem*, Trans. Amer. Math. Soc.**352**(2000), no. 6, 2825–2855. MR**1661305**, DOI 10.1090/S0002-9947-00-02472-7 - Dongming Wang,
*Computing triangular systems and regular systems*, J. Symbolic Comput.**30**(2000), no. 2, 221–236. MR**1777174**, DOI 10.1006/jsco.1999.0355 - F. R. Gantmacher,
*Théorie des matrices. Tome 1: Théorie générale*, Collection Universitaire de Mathématiques, No. 18, Dunod, Paris, 1966 (French). Traduit du Russe par Ch. Sarthou. MR**0225788** - Jean B. Lasserre,
*Polynomials with all zeros real and in a prescribed interval*, J. Algebraic Combin.**16**(2002), no. 3, 231–237 (2003). MR**1957101**, DOI 10.1023/A:1021848304877 - D. Lazard,
*Solving zero-dimensional algebraic systems*, J. Symbolic Comput.**13**(1992), no. 2, 117–131. MR**1153638**, DOI 10.1016/S0747-7171(08)80086-7 - F. Rouillier.
*Algorithmes efficaces pour l’étude des zéros réels des systèmes polynomiaux*, Ph.D. thesis (in French), Université de Rennes I, May 1996. - F. Rouillier. Private communication, 2002.
- Lieven Vandenberghe and Stephen Boyd,
*Semidefinite programming*, SIAM Rev.**38**(1996), no. 1, 49–95. MR**1379041**, DOI 10.1137/1038003

## Additional Information

**Jean B. Lasserre**- Affiliation: LAAS-CNRS and Institute of Mathematics, LAAS, 7 Avenue du Colonel Roche, 31077 Toulouse Cédex, France
- MR Author ID: 110545
- Email: lasserre@laas.fr
- Received by editor(s): April 10, 2002
- Published electronically: November 1, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**358**(2006), 1403-1420 - MSC (2000): Primary 12D10, 26C10, 30E05
- DOI: https://doi.org/10.1090/S0002-9947-05-03972-3
- MathSciNet review: 2186979