We introduce a novel application of Gröbner bases to solve (non-homogeneous) systems of integer linear equations over integers. For this purpose, we present a new algorithm which ascertains whether a linear system of equations has an integer solution or not; in the affirmative case, the general integer solution of the system is determined.

In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gröbner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I. P. Shestakov’s result that any Akivis algebra is linear and D. Segal’s result that the set of all good words in $X^{**}$ forms a linear basis of the free Pre-Lie algebra $\mathrm{PLie}\left(X\right)$ generated by the set $X$. For completeness,...

Let n, k, α be integers, n, α>0, p be a prime and q=p α. Consider the complete q-uniform family $$ℱ\left(k,q\right)=\left\{K\subseteq \left[n\right]:\left|K\right|\equiv k\left(modq\right)\right\}$$ We study certain inclusion matrices attached to F(k,q) over the field $${𝔽}_p$$ . We show that if l≤q−1 and 2l≤n then $$ran{k}_{{𝔽}_p}I(ℱ(k,q),\left(\begin{array}{c}\left[n\right]\\ ⩽\ell \end{array}\right))⩽\left(\begin{array}{c}n\\ \ell \end{array}\right)$$ This extends a theorem of Frankl [7] obtained for the case α=1. In the proof we use arguments involving Gröbner bases, standard monomials and reduction. As an application, we solve a problem of Babai and Frankl related to the size of some L-intersecting families modulo q.

For a monomial ideal I ⊂ S = K[x 1...,x n], we show that sdepth(S/I) ≥ n − g(I), where g(I) is the number of the minimal monomial generators of I. If I =νI′, where ν ∈ S is a monomial, then we see that sdepth(S/I) = sdepth(S/I′). We prove that if I is a monomial ideal I ⊂ S minimally generated by three monomials, then I and S/I satisfy the Stanley conjecture. Given a saturated monomial ideal I ⊂ K[x 1,x 2,x 3] we show that sdepth(I) = 2. As a consequence, sdepth(I) ≥ sdepth(K[x 1,x 2,x 3]//I) +1...

In this note we study some algebraic properties of Borel Ideals in the ring of polynomials over an effective field of characteristic zero by using a suitable partial order relation defined on the set of terms of each degree. In particular, in the three variable case, we characterize all the 0-dimensional Borel ideals corresponding to an admissible $h$-vector and their minimal free resolutions.