The purpose of this work is to extend the classification of planar point configurations with low Waldschmidt constants initiated in [9] and continued in [17] for all values less than 5/2. As a consequence we prove a conjecture of Dumnicki, Szemberg and Tutaj-Gasińska concerning initial sequences with low first differences.
We present a result on the generalized Hyers-Ulam stability of a functional equation in a single variable for functions that have values in a complete dislocated quasi-metric space. Next, we show how to apply it to prove stability of the Cauchy functional equation and the linear functional equation in two variables, also for functions taking values in a complete dislocated quasimetric space. In this way we generalize some earlier results proved for classical complete metric spaces.
We consider formal power series in several variables with coefficients in arbitrary field such that their Newton polyhedron has a loose edge. We show that if the symbolic restriction of the power series f to such an edge is a product of two coprime polynomials, then f factorizes in the ring of power series.
Let f and g be Weierstrass polynomials with coefficients in the ring of formal power series over an algebraically closed field of characteristic zero. Assume that f is irreducible and quasi-ordinary. We show that if degree of g is small enough and all monomials appearing in the resultant of f and g have orders big enough, then g is irreducible and quasi-ordinary, generalizing Abhyankar's irreducibility criterion for plane analytic curves.
We present a theorem about irreducibility of a polynomial that is the resultant of two others polynomials. The proof of this fact is based on the field theory. We also consider the converse theorem and some examples.
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