In this note we consider a perturbed mathematical programming problem where both the objective and the constraint functions are polynomial in all underlying decision variables and in the perturbation parameter ε. Recently, the theory of Gröbner bases was used to show that solutions of the system of first order optimality conditions can be represented as Puiseux series in ε in a neighbourhood of ε = 0. In this paper we show that the determination of the branching order and the order of the pole (if...

We generalize some results on reconstructing sets to the case of ideals of 𝕜[X₁,...,Xₙ]. We show that reconstructing sets can be approximated by finite subsets having the property of reconstructing automorphisms of bounded degree.

We extend results on reconstructing a polynomial automorphism from its restriction to the coordinate hyperplanes to some wider class of algebraic surfaces. We show that the algorithm proposed by M. Kwieciński in [K2] and based on Gröbner bases works also for this class of surfaces.

Given integers $D\>0,n\>1,0\<r\<n$ and a constant $C\>0$, consider the space of $r$-tuples $\overrightarrow{P}=(P_1...P_r)$ of real polynomials in $n$ variables of degree $\le D$, whose coefficients are $\le C$ in absolute value, and satisfying $\det {\left(\frac{\partial P_i}{\partial x_i}\left(0\right)\right)}_{1\le i,j\le r}=1$. We study the family $\left\{f\right|V\}$ of algebraic functions, where $f$ is a polynomial, and $V=\left\{\right|x|\le \delta ,\overrightarrow{P}(x)=0\},\delta \>0$ being a constant depending only on $n,D,C$. The main result is a quantitative extension theorem for these functions which is uniform in $\overrightarrow{P}$. This is used to prove Bernstein-type inequalities which are again uniform with respect to $\overrightarrow{P}$.The proof is based on...

When $S$ is a polynomial ring or more generally a standard graded algebra over a field $K$, with homogeneous maximal ideal $𝔪$, it is known that for an ideal $I$ of $S$, the regularity of powers of $I$ becomes eventually a linear function, i.e., $\mathrm{reg}\left(I^m\right)=dm+e$ for $m\gg 0$ and some integers $d$, $e$. This motivates writing $\mathrm{reg}\left(I^m\right)=dm+e_m$ for every $m\ge 0$. The sequence $e_m$, called the defect sequence of the ideal $I$, is the subject of much research and its nature is still widely unexplored. We know that $e_m$ is eventually constant. In this article, after...

We deal with a reduction of power series convergent in a polydisc with respect to a Gröbner basis of a polynomial ideal. The results are applied to proving that a Nash function whose graph is algebraic in a "large enough" polydisc, must be a polynomial. Moreover, we give an effective method for finding this polydisc.

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,...