Ideals and Quotients of Diagonally Quasi-Symmetric Functions

In 2004, J.-C. Aval, F. Bergeron and N. Bergeron studied the algebra of diagonally quasi-symmetric functions $\operatorname{\mathsf{DQSym}}$ in the ring $\mathbb{Q}[\mathbf{x},\mathbf{y}]$ with two sets of variables. They made conjectures on the structure of the quotient $\mathbb{Q}[\mathbf{x},\mathbf{y}]/\langle\operatorname{\mathsf{DQSym}}^+\rangle$, which is a quasi-symmetric analogue of the diagonal harmonic polynomials. In this paper, we construct a Hilbert basis for this quotient when there are infinitely many variables i.e. $\mathbf{x}=x_1,x_2,\dots$ and $\mathbf{y}=y_1,y_2,\dots$. Then we apply this construction to the case where there are finitely many variables, and compute the second column of its Hilbert matrix.

IDEALS OF POLYNOMIAL SEMIRINGS IN TROPICAL MATHEMATICS

We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from layered varieties, for which we prove that every prime ideal is a consequence of finitely many binomials. We also obtain layered tropical versions of the classical Principal Ideal Theorem and Hilbert Basis Theorem.

On infinitely cohomologous to zero observables

We show that for a large class of piecewise expanding maps T, the bounded p-variation observables u0 that admit an infinite sequence of bounded p-variation observables ui satisfying \[ u_{i}= u_{i+1}\circ T-u_{i+1} \] are constant. The method of the proof consists of finding a suitable Hilbert basis for L2(hm), where hm is the unique absolutely continuous invariant probability of T. On this basis, the action of the Perron–Frobenius and the Koopman operator on L2(hm) can be easily understood. This result generalizes earlier results by Bamón, Kiwi, Rivera-Letelier and Urzúa for the case T(x)=ℓx mod 1 , ℓ∈ℕ∖ {0,1} and Lipschitzian observables u0.

Noetherian orders

Noether classes of posets arise in a natural way from the constructively meaningful variants of the notion of a Noetherian ring. Using an axiomatic characterisation of a Noether class, we prove that if a poset belongs to a Noether class, then so does the poset of the finite descending chains. When applied to the poset of finitely generated ideals of a ring, this helps towards a unified constructive proof of the Hilbert basis theorem for all Noether classes.