### A note on Babylonian square-root algorithm and related variants.

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Let $k$ be a field. We compute the set ${\left[{\mathbf{P}}^{1},{\mathbf{P}}^{1}\right]}^{\mathrm{N}}$ ofnaivehomotopy classes of pointed $k$-scheme endomorphisms of the projective line ${\mathbf{P}}^{1}$. Our result compares well with Morel’s computation in [11] of thegroup${\left[{\mathbf{P}}^{1},{\mathbf{P}}^{1}\right]}^{{\mathbf{A}}^{1}}$ of ${\mathbf{A}}^{1}$-homotopy classes of pointed endomorphisms of ${\mathbf{P}}^{1}$: the set ${\left[{\mathbf{P}}^{1},{\mathbf{P}}^{1}\right]}^{\mathrm{N}}$ admits an a priori monoid structure such that the canonical map ${\left[{\mathbf{P}}^{1},{\mathbf{P}}^{1}\right]}^{\mathrm{N}}\to {\left[{\mathbf{P}}^{1},{\mathbf{P}}^{1}\right]}^{{\mathbf{A}}^{1}}$ is a group completion.

Using the principle that characteristic polynomials of matrices obtained from elements of a reductive group $\mathbf{G}$ over $\mathbf{Q}$ typically have splitting field with Galois group isomorphic to the Weyl group of $\mathbf{G}$, we construct an explicit monic integral polynomial of degree $240$ whose splitting field has Galois group the Weyl group of the exceptional group of type ${\mathbf{E}}_{8}$.

Border bases are an alternative to Gröbner bases. The former have several more desirable properties. In this paper some constructions and operations on border bases are presented. Namely; the case of a restriction of an ideal to a polynomial ring (in a smaller number of variables), the case of the intersection of two ideals, and the case of the kernel of a homomorphism of polynomial rings. These constructions are applied to the ideal of relations and to factorizable derivations.

Let ${f}_{i}$ be polynomials in $n$ variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials ${g}_{i}$ such that $\sum {g}_{i}{f}_{i}=1$. The effective versions of this result bound the degrees of the ${g}_{i}$ in terms of the degrees of the ${f}_{j}$. The aim of this paper is to generalize this to the case when the ${f}_{i}$ are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.

In this paper a new method which is a generalization of the Ehrlich-Kjurkchiev method is developed. The method allows to find simultaneously all roots of the algebraic equation in the case when the roots are supposed to be multiple with known multiplicities. The offered generalization does not demand calculation of derivatives of order higher than first simultaneously keeping quaternary rate of convergence which makes this method suitable for application from practical point of view.

We study infinite translation surfaces which are $\mathbb{Z}$-covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition for recurrence of their straight-line flows. Extending results of Hubert and Schmithüsen, we provide examples of infinite non-arithmetic lattice surfaces, as well as surfaces with infinitely generated Veech groups.