Subalgebra analogue of Standard bases for ideals in $ K[[t_{1}, t_{2}, \ldots, t_{m}]][x_{1}, x_{2}, \ldots, x_{n}] $

<abstract><p>In this paper, we develop a theory for Standard bases of $ K $-subalgebras in $ K[[t_{1}, t_{2}, \ldots, t_{m}]] [x_{1}, x_{2}, ..., x_{n}] $ over a field $ K $ with respect to a monomial ordering which is local on $ t $ variables and we call them Subalgebra Standard bases. We give an algorithm to compute subalgebra homogeneous normal form and an algorithm to compute weak subalgebra normal form which we use to develop an algorithm to construct Subalgebra Standard bases. Throughout this paper, we assume that subalgebras are finitely generated.</p></abstract>

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].

On multivariate polynomials with many roots over a finite grid

In this paper, we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [Footprints or generalized Bezout’s theorem, IEEE Trans. Inform. Theory 46(2) (2000) 635–641, On (or in) Dick Blahut’s ‘footprint’, Codes[Formula: see text] Curves Signals (1998) 3–9] provides us with an upper bound on the number of roots, and this bound is sharp in that it can always be attained by trivial polynomials being a constant times a product of an appropriate combination of terms consisting of a variable minus a constant. In contrast to the one variable case, there are multivariate polynomials attaining the footprint bound being not of the above form. This even includes irreducible polynomials. The purpose of the paper is to determine a large class of polynomials for which only the mentioned trivial polynomials can attain the bound, implying that to search for other polynomials with the maximal number of roots one must look outside this class.

Computation of minimal homogeneous generating sets and minimal standard bases for ideals of free algebras

Let {K\langle X\rangle=K\langle X_{1},\ldots,X_{n}\rangle} be the free algebra generated by {X=\{X_{1},\ldots,X_{n}\}} over a field K. It is shown that, with respect to any weighted {\mathbb{N}}-gradation attached to {K\langle X\rangle}, minimal homogeneous generating sets for finitely generated graded two-sided ideals of {K\langle X\rangle} can be algorithmically computed, and that if an ungraded two-sided ideal I of {K\langle X\rangle} has a finite Gröbner basis {{\mathcal{G}}} with respect to a graded monomial ordering on {K\langle X\rangle}, then a minimal standard basis for I can be computed via computing a minimal homogeneous generating set of the associated graded ideal {\langle\mathbf{LH}(I)\rangle}.

Recognizing the Semiprimitivity of ℕ-graded Algebras via Gröbner Bases

Let K〈X〉=K〈X1,…,Xn〉 be the free K-algebra on X={X1,…,Xn} over a field K, which is equipped with a weight ℕ-gradation (i.e., each Xi is assigned a positive degree), and let [Formula: see text] be a finite homogeneous Gröbner basis for the ideal [Formula: see text] of K〈X〉 with respect to some monomial ordering ≺ on K〈X〉. It is shown that if the monomial algebra [Formula: see text] is semiprime, where [Formula: see text] is the set of leading monomials of [Formula: see text] with respect to ≺, then the ℕ-graded algebra A=K〈X〉 /I is semiprimitive in the sense of Jacobson. In the case that [Formula: see text] is a finite nonhomogeneous Gröbner basis with respect to a graded monomial ordering ≺ gr , and the ℕ-filtration FA of the algebra A=K〈X〉 /I induced by the ℕ-grading filtration FK〈X〉 of K〈X〉 is considered, if the monomial algebra [Formula: see text] is semiprime, then it is shown that the associated ℕ-graded algebra G(A) and the Rees algebra à of A determined by FA are all semiprimitive.

A note on multivariate polynomial division and Gröbner bases

We first present purely combinatorial proofs of two facts: the well-known fact that a monomial ordering must be a well ordering, and the fact (obtained earlier by Buchberger, but not widely known) that the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not the leading term, but is freely chosen. The latter is then used to introduce a previously unnoted, seemingly weaker, criterion for an ideal basis to be Grobner, and to suggest a new heuristic approach to Grobner basis computations.

NUMERICAL ALGORITHMS FOR DUAL BASES OF POSITIVE-DIMENSIONAL IDEALS

An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However the usual standard basis algorithms are not numerically stable. A numerically stable approach to describing the ideal is by finding the space of dual functionals that annihilate it, which reduces the problem to one of linear algebra. There are several known algorithms for finding the truncated dual up to any specified degree, which is useful for describing zero-dimensional ideals. We present a stopping criterion for positive-dimensional cases based on homogenization that guarantees all generators of the initial monomial ideal are found. This has applications for calculating Hilbert functions.

POLYNOMIAL KNOT AND LINK INVARIANTS FROM THE VIRTUAL BIQUANDLE

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek polynomial) for virtual knots and the classical Alexander polynomial for classical knots. For a fixed monomial ordering <, the Gröbner bases for these ideals are computable, comparable invariants which fully determine the elementary ideals and which generalize and unify the classical and generalized Alexander polynomials. We provide examples to illustrate the usefulness of these invariants and propose questions for future work.