On the real zeros of positive semi-definite biquadratic forms
نویسندگان
چکیده
For a positive semi-definite biquadratic forms F in (3, 3) variables, we prove that if F has a finite number but at least 7 real zeros Z(F ), then it is not a sum of squares. We show also that if F has at least 11 zeros, then it has infinitely many real zeros and it is a sum of squares. It can be seen as the counterpart for biquadratic forms as the results of Choi, Lam and Reznick for positive semi-definite ternary sextics. We introduce and compute some of the numbers BBn,m which are set to be equal to sup |Z(F )| where F ranges over all the positive semi-definite biquadratic forms F in (n,m) variables such that |Z(F )| < ∞. We also recall some old constructions of positive semi-definite biquadratic forms which are not sums of squares and we give some new families of examples.
منابع مشابه
Sur Une Classe De Formes Biquadratiques Semi-définies Positives * on a Class of Positive Semidefinite Biquadratic Forms
Continuing the study of positive semidefinite biquadratic forms on Rm × Rn ([1], [9] and [10]), we characterize those among them that are the sum of squares of bilinear forms.
متن کاملPiecewise Certificates of Positivity for matrix polynomials
We show that any symmetric positive definite homogeneous matrix polynomial M ∈ R[x1, . . . , xn] admits a piecewise semi-certificate, i.e. a collection of identites M(x) = P j fi,j(x)Ui,j(x) T Ui,j(x) where Ui,j(x) is a matrix polynomial and fi,j(x) is a non negative polynomial on a semialgebraic subset Si, where R = ∪ri=1Si. This result generalizes to the setting of biforms. Some examples of c...
متن کاملSur une classe de formes biquadratiques semi-definies positives
Continuing the study of positive semidefinite biquadratic forms on Rm × Rn ([1], [9] and [10]), we characterize those among them that are the sum of squares of bilinear forms.
متن کاملDomain of attraction of normal law and zeros of random polynomials
Let$ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraicpolynomial, where $A_{0},A_{1}, cdots $ is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus $A_j$'s for $j=0,1cdots $ possesses the characteristic functions $exp {-frac{1}{2}t^{2}H_{j}(t)}$, where $H_j(t)$'s are complex slowlyvarying functions.Under the assumption that there exist ...
متن کاملOn Positive Definite Solutions of Quaternionic Matrix Equations
The real representation of the quaternionic matrix is definited and studied. The relations between the positive (semi)define quaternionic matrix and its real representation matrix are presented. By means of the real representation, the relation between the positive (semi)definite solutions of quaternionic matrix equations and those of corresponding real matrix equations is established. Keywords...
متن کامل