A procedure is given to determine whether or not a polynomial in several real variables is globally positive. Emphasis is placed on the resolution of singularities of various types that occur during the implementation of the algorithm. A nontrivial example encountered in application of Lyapunov direct method to stability investigation of nonlinear control systems is given to illustrate the details of the procedure.

General procedure for multivariable polynomial positivity test with control applications

A positivity method for forms of even order is reported. It is shown that testing for positivity of such forms can be reduced to a test for positive definiteness of parametric-matrices. Special emphasis is placed on quartic- and sixth-order forms. Examples are given to illustrate the proposed criterion.

A procedure for the positive definiteness of forms of even order

The notion of the inners of a matrix is fully discussed. The inners applications to control theory, stability theory, communication theory, circuit theory, network theory, digital filters, bioengineering, sparse matrix theory, quantum physics, and some topics in mathematics are enumerated and analyzed. It is shown that the inners concept offers a theoretical as well as computational unification for these applications. In addition, the historical background and motivation is presented for the inners approach. The importance of the inners notion to education, computation, and research in system theory is surveyed and evaluated. Future research problems using this concept are enumerated. Finally, this survey is documented by many past and recent references.

The theory and applications of the inners

A procedure for resolution of the positivity and the nonnegativity of a real polynomial in n real variables is described. The procedure is based upon an augmented theory of resultants. The sufficiency condition, when invoked using exact arithmetic, gives an unambiguous (absolutely certain) result. Several examples are provided and comments on the attributes of this and other (alternative) procedures are given.

http://ieeexplore.ieee.org/iel5/31/23484/01084545.pdf

Real polynomials: Nonnegativity and positivity

The role of elementary decision algebra in the broad area of multidimensional problems is given, with emphasis on algorithms and computational simplification. The encouraging fact that several difficult problems can be solved in a finite number of steps via rational operations opens up unlimited scope for research into the search for more efficient algorithms. It is pointed out via a counterexample that the only existing synthesis procedure for arbitrary passive multiports with prescribed multivariable positive real matrices is incorrect. In the process, several links to the study of positivity preserving linear transformations which map square matrices into square matrices are unravelled.

New techniques and results in multidimensional problems

It is shown how the test for positive definiteness of an arbitrary binary form can be expressed in terms of an inner algorithm. An example is used to illustrate the computational aspects of the procedure.

https://ieeexplore.ieee.org/iel5/9/24146/01100875.pdf

E. Jury

Papers

Inner algorithm to test for positive definiteness of arbitrary binary forms