Singularities of type \(A_k\) on plane curves of a chosen degree (Q5930961)

There is a general problem to describe singularities that can occur on algebraic hypersurfaces. The authors study a version of this problem for plane curves of a chosen degree. They show an upper bound \((d-1)^2-[d/2]([d/2]-1)\geq k(d)\) to the maximal number \(k(d)\) such that there is a plane curve of degree \(d\) with singularity \(A_k\). The technics developed by the authors have many interesting applications, especially they prove that the uniform Łojasiewicz exponent \(M_d\) for polynomials of degree \(d\) in two variables is bounded from below by \((15/28)d^2+O(d)\). This completes the estimate \(M_d\leq(d-1)^2+1\), which is due to J. Gwoździewicz.