Methods from computer algebra, mostly so called Grr obner bases from commutative algebra, are used to solve the algebraic Riccati equation (ARE) symbolically. The methods suggested allow us to track the innuence of parameters in the system or penalty matrices on the solution. Some non-trivial aspects arise when addressing the problem from the point of view commutative algebra, for example the o...

Let K be an infinite field and K〈X〉 = K〈X1, ..., Xn〉 the free associative algebra generated by X = {X1, ..., Xn} over K. It is proved that if I is a two-sided ideal of K〈X〉 such that the K-algebra A = K〈X〉/I is almost commutative in the sense of [3], namely, with respect to its standard N-filtration FA, the associated N-graded algebra G(A) is commutative, then I is generated by a finite Gröbner...

We build, using the notion of zinbiel algebra, some commutative subalgebras C_{u,v} inside an algebra formal iterated integrals. There is a quotient map from this integrals to motivic multiple zeta values. Restricting gives morphism graded algebras with same dimension. This conjectured be generically isomorphism. When u+v = 0, image instead sub-algebra