It is often a good idea to look at the coefficients of the polynomial from a number theoretical standpoint. The general principle is that any polynomial equation can be reduced mod m to obtain another polynomial equation whose coefficients are the residue classes mod m. Many criterions exist for testing whether a polynomial is irreducible. Unfortunately, none are powerful enough to be universal...

We present a powerful computational method for automatically generating polynomial invariants of hybrid systems with linear continuous dynamics. When restricted to linear continuous dynamical systems, our method generates a set of polynomial equations (algebraic set) that is the best such over-approximation of the reach set. This shows that the set of algebraic invariants of a linear system is ...

We consider symmetric (as well as multi-symmetric) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of fixed degrees. We give polynomial (in the dimension of the ambient space) bounds on the number of irreducible representations of the symmetric group which acts on these sets, as well as their multip...

Differing from [6] this paper reduces non-linear ranking function discovering for polynomial programs to semi-algebraic system solving, and demonstrates how to apply the symbolic computation tools, DISCOVERER and QEPCAD, to some interesting examples. keywords: Program Verification, Loop Termination, Ranking Function, Polynomial Programs, Semi-Algebraic Systems, Computer Algebra, DISCOVERER, QEPCAD

This paper extends the modified affine arithmetic in matrix form method for bivariate polynomial evaluation and algebraic curve plotting in 2D to modified affine arithmetic in tensor form for trivariate polynomial evaluation and algebraic surface plotting in 3D. Experimental comparison shows that modified affine arithmetic in tensor form is not only more accurate but also much faster than affin...

We analyze Glad/Fliess algebraic observability for polynomial control systems from a commutative algebraic/algebro-geometric point of view, using some results from the theory of Grr obner bases. Furthermore, we discuss some topics in realization theory for polynomial diierential equations. Most issues are treated in a constructive framework.

i=0 ai,kv i = 0. Recall that, since v is trancendental over K, the set {1, v, v, ...} is linearly independent over K. This implies that ai,k = 0 for all k > 0 (otherwise there would be a finite linear combination of powers of v which is zero, hence v would be algebraic over K). Thus each hi above is of the form hi(x) = ai,0 (a constant polynomial), namely hi ∈ K. But then (2) implies that v is ...

We introduce a class of dynamical systems of algebraic origin, consisting of selfinteracting irreducible polynomials over a field. A polynomial f is made to act on a polynomial g by mapping the roots of g. This action identifies a new polynomial h, as the minimal polynomial of the displaced roots. By allowing several polynomials to act on one another, we obtain a self-interacting system with a ...