On bisecants of R\'edei type blocking sets and applications

We use polynomial techniques to derive structural results on R'edei type blocking sets from information on their bisecants. We apply our results to point sets of $PG(2,q)$ with few odd-secants. In particular, we improve the lower bound of Balister, Bollob'as, F"uredi and Thompson on the number of odd-secants of a $(q+2)$-set in $PG(2,q)$ and we answer a related open question of Vandendriessche. We prove structural results for semiovals and derive the non existence of semiovals of size $q+3$ when 3 does not divide $q$ and $q>5$. This extends a result of Blokhuis who classified semiovals of size $q+2$, and a result of Bartoli who classified semiovals of size $q+3$ when $q\le 17$. In the $q$ even case we can say more applying a result of Sz\H{o}nyi and Weiner about the stability of sets of even type. We also obtain new proof to a result of G'acs and Weiner about $(q+t,t)$-arcs of type $(0,2,t)$ and to one part of a result of Ball, Blokhuis, Brouwer, Storme and Sz\H{o}nyi about functions over $GF(q)$ determining less than $(q+3)\mathrm{/}2$ directions.