  An alternative to Plemelj - Smithies formulas for the p -regularized
quantities $d^{(p)}(K)$ and $D^{(p)}(K)$ is presented which generalizes
previous expressions with $p=1$ due to Grothendieck and Fredholm. It is also
presented global upper bounds for these quantities. In particular we prove that
$$ |d^{(p)}(K)| \le e^{\kappa \n K \n_p^p} $$ holds with $\kappa = \kappa(p)
\le \kappa(\infty) = \exp \bigl\{-{1 \over 4(1+ e^{2 \pi})} \bigr\}$ for $ p\ge
3$ which improves previous estimate yielding $\kappa(p) = e (2 + \ln(p-1))$.
