A zero-dimensional polynomial ideal may have a lot of complex zeros. But sometimes, only some of them are needed. In this paper, for a zero-dimensional ideal I , we study its complex zeros that locate in another variety V(J) where J is an arbitrary ideal. The main problem is that for a point in V(I) ∩ V(J) = V(I + J), its multiplicities w.r.t. I and I + J may be different. Therefore, we cannot ...

The aim of this paper is to prove the stability in the sense of Hyers–Ulam stability of a polynomial equation. More precisely, if x is an approximate solution of the equation x + αx + β = 0, then there exists an exact solution of the equation near to x.

In this paper, operational matrix method based upon the Bernstein polynomials is proposed to solve the variable order fractional integro-differential equations in the Caputo derivative sense. We derive the Bernstein polynomials operational matrix of fractional order integration and introduce the product operational matrix of Bernstein polynomials. A truncated the Bernstein polynomials series to...

In this paper we investigate the Boolean functions with essential arity gap 2. We use Full Conjunctive Normal Forms instead of Zhegalkin’s polynomials, which allow us to simplify the proofs and to obtain several combinatorial results, concerning the Boolean functions with a given arity gap.

Inverse polynomial images of [−1, 1], which consists of two Jordan arcs, are characterised by an explicit polynomial equation for the four endpoints of the arcs. MSC: 41A10; 30C10

We present short elementary proofs of the well-known Ruffini-Abel-Galois theorems on unsolvability of algebraic equations in radicals. This proof is obtained from existing expositions by stripping away material not required for the proof (but presumably required elsewhere). In particular, we do not use the terms ‘Galois group’ and even ‘group’. However, our presentation is a good way to learn (...

We factor the virtual Poincaré polynomial of every homogeneous space G/H , where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t(t − 1)QG/H(t ) for a polynomial QG/H with non-negative integer coefficients. Moreover, we show that QG/H(t ) divides the virtual Poincaré polynomial of every regular embedding of G/H , if H is connected. Introduction and statement ...

We prove that for any set E ⊆ Z with upper Banach density d(E) > 0, the set “of cubic configurations” in E is large in the following sense: for any k ∈ N and any ε > 0, the set {(n1, . . . , nk) ∈ Z k : d( ⋂ e1,...,ek∈{0,1} (E − (e1n1 + · · ·+ eknk))) > d (E) k − ε} is an AVIP0set. We then generalize this result to the case “of polynomial cubic configurations” e1p1(n) + · · · + ekpk(n) where th...

Proof. Let f1, . . . , fm be generators of A(X) over A(Y ). Then the fi also generate K(X) over K(Y ), so we can reorder indices such that f1, . . . , fr are algebraically independent over K(Y ). Let R = A(Y )[f1, . . . , fr] ⊆ A(X). Since the fi are algebraically independent over K(Y ) they are algebraically independent over A(Y ), so R is isomorphic to an r-variable polynomial ring, which is ...