arXiv subjects
Publications and source records attributed to Beata Gryszka.
Let $x=t^n$, $y=\sum_{i=1}^{\infty}a_it^i$ be a parametrisation of the germ of a complex plane analytic curve $Γ$ at the origin. Then $Γ$ has the implicit equation $f(x,y)=0$ in the neighbourhood of the origin, where $f=\sum c_{ij}x^iy^j$ is a Weierstrass polynomial in $\mathbb{C}[[x]][y]$ of degree $n$. Every polynomial depending on coefficients $c_{ij}$ can be expressed as a polynomial depending on the coefficients $a_i$.
Let $\mathbb{K}$ be an uncountable field of characteristic zero and let $f$ be a function from $\mathbb{K}^n$ to $\mathbb{K}$. We show that if the restriction of $f$ to every affine plane $L\subset\mathbb{K}^n$ is regular, then $f$ is a regular function.
Let $(f,g)\colon (\mathbb{C}^2,0)\longrightarrow (\mathbb{C}^2,0)$ be a holomorphic mapping with an isolated zero. We show that the initial Newton polynomial of its discriminant is determined, up to rescalling variables, by the ideals $(f)$ and $(g)$.