A Moment Approach to Analyze Zeros of Triangular Polynomial Sets
نویسنده
چکیده
Let I = 〈g1, . . . , gn〉 be a zero-dimensional ideal of R[x1, . . . , xn] such that its associated set G of polynomial equations gi(x) = 0 for all i = 1, . . . , n is in triangular form. By introducing multivariate Newton sums we provide a numerical characterization of polynomials in √ I. We also provide a necessary and sufficient (numerical) condition for all the zeros of G to be in a given set K ⊂ Cn, without explicitly computing the zeros. In addition, we also provide a necessary and sufficient condition on the coefficients of the gi’s for G to have (a) only real zeros, (b) to have only real zeros, all contained in a given semi-algebraic set K ⊂ Rn. In the proof technique, we use a deep result of Curto and Fialkow (2000) on the 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 gi’s via the Newton sums of G. In addition, the number of distinct real zeros is shown to be the maximal rank of a related moment matrix.
منابع مشابه
A pr 2 00 2 Transfer Matrices and Partition - Function Zeros for Antiferromagnetic Potts Models III . Triangular - lattice chromatic polynomial
We study the chromatic polynomial PG(q) for m×n triangular-lattice strips of widths m ≤ 12P, 9F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such...
متن کاملA pr 2 00 3 Transfer Matrices and Partition - Function Zeros for Antiferromagnetic Potts Models III . Triangular - lattice chromatic polynomial
We study the chromatic polynomial PG(q) for m×n triangular-lattice strips of widths m ≤ 12P, 9F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such...
متن کاملInequalities for the polar derivative of a polynomial with $S$-fold zeros at the origin
Let $p(z)$ be a polynomial of degree $n$ and for a complex number $alpha$, let $D_{alpha}p(z)=np(z)+(alpha-z)p'(z)$ denote the polar derivative of the polynomial p(z) with respect to $alpha$. Dewan et al proved that if $p(z)$ has all its zeros in $|z| leq k, (kleq 1),$ with $s$-fold zeros at the origin then for every $alphainmathbb{C}$ with $|alpha|geq k$, begin{align*} max_{|z|=...
متن کاملOn the Theories of Triangular Sets
Triangular sets appear under various names in many papers concerning systems of polynomial equations. Ritt (1932) introduced them as characteristic sets. He described also an algorithm for solving polynomial systems by computing characteristic sets of prime ideals and factorizing in field extensions. Characteristic sets of prime ideals have good properties but factorization in algebraic extensi...
متن کاملOn the polar derivative of a polynomial
For a polynomial p(z) of degree n, having all zeros in |z|< k, k< 1, Dewan et al [K. K. Dewan, N. Singh and A. Mir, Extension of some polynomial inequalities to the polar derivative, J. Math. Anal. Appl. 352 (2009) 807-815] obtained inequality between the polar derivative of p(z) and maximum modulus of p(z). In this paper we improve and extend the above inequality. Our result generalizes certai...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2005