arXiv subjects
Publications and source records attributed to Cezar Joita.
We study the bifurcation values of real polynomial maps $f: \bR^{2n} \to \bR^2$ which reflect the lack of asymptotic regularity at infinity. We formulate real counterparts of some structure results which have been previously proved in case of complex polynomials by Kushnirenko, Némethi and Zaharia and other authors, emphasizing the typical real phenomena that occur.
The problem of detecting the bifurcation set of polynomial mappings $\mathbb{ C}^m \to \mathbb{ C}^k$, $m\ge 2$, $m\ge k\ge 1$, has been solved in the case $m=2$, $k=1$ only. Its solution, which goes back to the 1970s, involves the non-constancy of the Euler characteristic of fibres. We provide a complete answer to the general case $m= k+1 \ge 3$ in terms of the Betti numbers of fibres and of a vanishing phenomenon discovered in the late 1990s in the real setting.
We study the third Cauchy-Fantappie formula, an integral representation formula for holomorphic functions on a domain in affine space, presented by Jean Leray in the third paper of his famous series "Probleme de Cauchy", published in 1959. We show by means of examples that this formula does not hold without some additional conditions, left unmentioned by Leray. We give sufficient conditions and a necessary condition for the formula to hold, and, in the case of a contractible domain, characterize it cohomologically.