On Finite Noncommutative Gröbner Bases

It is well known that in the noncommutative polynomial ring in serveral variables Buchberger’s algorithm does not always terminate. Thus, it is important to characterize noncommutative ideals that admit a finite Gröbner basis. In this context, Eisenbud, Peeva and Sturmfels defined a map γ from the noncommutative polynomial ring k⟨X1, …, Xn⟩ to the commutative one k[x1, …, xn] and proved that any ideal [Formula: see text] of k⟨X1, …, Xn⟩, written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn], amits a finite Gröbner basis with respect to a special monomial ordering on k⟨X1, …, Xn⟩. In this work, we approach the opposite problem. We prove that under some conditions, any ideal [Formula: see text] of k⟨X1, …, Xn⟩ admitting a finite Gröbner basis can be written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn].

Factorization of Noncommutative Polynomials and Nullstellensätze for the Free Algebra

This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots ,x_g)$ is $\mathscr{Z}(\,f)=(\mathscr{Z}_n(\,f))_n$, where $\mathscr{Z}_n(\,f)=\{X \in{\operatorname{M}}_{n}({\mathbb{C}})^g \colon \det f(X) = 0\}.$ The 1st main theorem of this article shows that the irreducible factors of $f$ are in a natural bijective correspondence with irreducible components of $\mathscr{Z}_n(\,f)$ for every sufficiently large $n$. With each polynomial $h$ in $x$ and $x^*$ one also associates its real singularity set $\mathscr{Z}^{{\operatorname{re}}}(h)=\{X\colon \det h(X,X^*)=0\}$. A polynomial $f$ that depends on $x$ alone (no $x^*$ variables) will be called analytic. The main Nullstellensatz proved here is as follows: for analytic $f$ but for $h$ dependent on possibly both $x$ and $x^*$, the containment $\mathscr{Z}(\,f) \subseteq \mathscr{Z}^{{\operatorname{re}}} (h)$ is equivalent to each factor of $f$ being “stably associated” to a factor of $h$ or of $h^*$. For perspective, classical Hilbert-type Nullstellensätze typically apply only to analytic polynomials $f,h $, while real Nullstellensätze typically require adjusting the functions by sums of squares of polynomials (sos). Since the above “algebraic certificate” does not involve a sos, it seems justified to think of this as the natural determinantal Hilbert Nullstellensatz. An earlier paper of the authors (Adv. Math. 331 (2018): 589–626) obtained such a theorem for special classes of analytic polynomials $f$ and $h$. This paper requires few hypotheses and hopefully brings this type of Nullstellensatz to near final form. Finally, the paper gives a Nullstellensatz for zeros ${\mathcal{V}}(\,f)=\{X\colon f(X,X^*)=0\}$ of a hermitian polynomial $f$, leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable.

Rooted tree maps and the derivation relation for multiple zeta values

Rooted tree maps assign to an element of the Connes–Kreimer Hopf algebra of rooted trees a linear map on the noncommutative polynomial algebra in two letters. Evaluated at any admissible word, these maps induce linear relations between multiple zeta values. In this note, we show that the derivation relations for multiple zeta values are contained in this class of linear relations.

Real nullstellensatz and *-ideals in *-algebras

For a fixed tuple of square matrices X ={X_1,...,X_g} the set I(X) of all noncommutative polynomials p in X and X∗ such that p(X) = 0 is an ideal in the ∗-algebra of all polynomials. This article concerns such zeroes and their corresponding ideals. An algebraic characterization of ideals of the form I(X) is a real nullstellensatz. A main result of this article is a strong nullstellensatz for a ∗-ideal of finite codimension in a ∗-algebra. Without the finite codimension assumption, there are examples of such ideals which do not satisfy, very liberally interpreted, any Nullstellensatz. A polynomial p in noncommuting variables (x_1,...,x_g,x∗_1,...,x_∗g) is called analytic if it is a polynomial in the variables x_j only. As shown in this article, ∗-ideals generated by analytic polyno-mials do satisfy a natural Nullstellensatz and those generated by homogeneous analytic polynomials have a particularly simple description. Another natural notion of zero of a noncommutative polynomial p is a pair (X, v) such that p(X)v = 0; here X is an n by n matrix tuple and v ∈ R^n. For fixed (X,v), the set of all such polynomials is a left ideal. The relationship between such zeroes and their left ideals is considerably more developed than is our beginning effort here. This article provides a guide to that literature.