Rank constraints on matrices emerge in many automatic control applications. In this short document we discuss how to rewrite the constraint into a polynomial equations of the elements in a the matrix. If addition semidefinite matrix constraints are included, the polynomial equations can be turned into an inequality. We also briefly discuss how to implement these polynomial constraints.

In the series of papers [SS93a, SS93b, SS93c, SS94, SS96], Shub and Smale defined and studied in depth a homotopy method for solving systems of polynomial equations. Some articles preceding this new treatment were [Kan49, Sma86, Ren87, Kim89, Shu93]. Other authors have also treated this approach in [Mal94, Yak95, Ded97, BCSS98, Ded01, Ded06, MR02] and more recently in [Shu08, BS08]. In a previo...

A worst case bound for the condition number of a generic system of polynomial equations with integer coefficients is given. For fixed degree and number of equations, the condition number is (non-uniformly, generically) pseudo-polynomial in the input size.

the main objective of present research work is to formulate the clopidogrel fast dissolving tablets. clopidogrel, an antiplatelet drug, belongs to bcs class-ii and used to control heart attack, hypertension by inhibiting platelet activation and aggregation .the fast dissolving tablets of clopidogrel were prepared employing different concentrations of crospovidone and croscarmellose sodium in di...

Abstract Kumar et al. (2006) obtained a fifth order polynomial in ω for the dispersion relation and pointed out that the calculations preformed by Porter et al. (1994) and by Dwivedi & Pandey (2003) seem to be in error, as they obtained a sixth order polynomial. The energy equation of Dwivedi & Pandey (2003) was dimensionally wrong. Dwivedi & Pandey (2006) corrected the energy equation and stil...

We study real-polynomial solutions P (x) of difference equations of the formG(P (x−τ1), . . . , P (x− τs)) +G0(x)=0, where τi are real numbers, G(x1, . . . , xs) is a real polynomial of a total degree D ≥ 2, and G0(x) is a polynomial in x. We consider the following problem: given τi, G and G0, find an upper bound on the degree d of a real-polynomial solution P (x), if exists. We reduce this pro...

Systems of polynomial equations arise naturally in applications ranging from the study of chemical reactions to coding theory to geometry and number theory. Furthermore, the complexity of the equations we wish to solve continues to rise: while engineers in ancient Egypt needed to solve quadratic equations in one variable, today we have applications in satellite orbit design and combustive fluid...

Lie-algebraic approach to the theory of polynomial solutions. II. Differential equations in one real and one Grassmann variables and 2x2 matrix differential equations ABSTRACT A classification theorem for linear differential equations in two variables (one real and one Grassmann) having polynomial solu-tions(the generalized Bochner problem) is given. The main result is based on the consideratio...

Smale’s α-theory certifies that Newton iterations will converge quadratically to a solution of a square system of analytic functions based on the Newton residual and all higher order derivatives at the given point. Shub and Smale presented a bound for the higher order derivatives of a system of polynomial equations based in part on the degrees of the equations. For a given system of polynomial-...