Abstract: We study the image of an additive polynomial $f$ over a field $k$ of characteristic $p > 0$. We define the additive rank of $f$ over $k$ to be the smallest positive integer $r$ such that there exists an additive polynomial $g$ in $r$ variables with coefficients in $k$ which generates the same image as $f$ does. We show that over perfect fields the additive ranks of (non-zero) additive polynomials are always 1. We also show that for every positive integer $r$, there is some additive polynomial over a certain field with additive rank $r$.

Abstract: In this talk first I will give some examples of vector bundles on $S^2$ which appears to be very important in complex topological K-theory and show how to imagine them. Then I will define clutching map of vector bundles on $S^n$ which will be used to prove the Bott periodicity. Then we are prepared to give a definition of K-group and ring structure on it. At last is the statement of Bott periodicity which I will prove in the next week.