Continuous functions on the plane regular after one blowing-up (Q509996)

In the paper under review, the authors study rational functions admitting a continuous extension to the real affine space. Following the terminology introduced by \textit{G. Fichou} et al. [J. Reine Angew. Math. 718, 103--151 (2016; Zbl 1390.14172)], the authors call the class of continuous functions lying between rational and regular functions regulous, and address two questions relevant to this class: firstly, they investigate regularity of such functions exhibiting some nice properties of their partial derivatives, and secondly, since these functions correspond to rational functions which become regular after some blowings-up, they work on the plane where it suffices to blow-up points and count the number of stages of necessary blowings-up. The authors then investigate the ring of rational continuous functions on the plane regular after one stage of blowings-up. In particular, a version of Positivstellensatz without denominator is shown to hold in this ring.