Let V, W be algebraic subsets of $k^n$, $k^m$ respectively, with n ≤ m. It is known that any finite polynomial mapping f: V → W can be extended to a finite polynomial mapping $F:k^n\to k^m.$ The main goal of this paper is to estimate from above the geometric degree of a finite extension $F:k^n\to k^n$ of a dominating mapping f: V → W, where V and W are smooth algebraic sets.

For $1\le c\le p-1$, let $E_1,E_2,\cdots ,E_m$ be fixed numbers of the set $\{0,1\}$, and let $a_1,a_2,\cdots ,a_m$ $(1\le a_i\le p$, $i=1,2,\cdots ,m)$ be of opposite parity with $E_1,E_2,\cdots ,E_m$ respectively such that $a_1{a}_2\cdots a_m\equiv c(modp)$. Let $$N(c,m,p)=\frac{1}{2^{m-1}}\underset{a_1{a}_2\cdots a_m\equiv c(modp)}{\sum _{a_1=1}^{p-1}\sum _{a_2=1}^{p-1}\cdots \sum _{a_m=1}^{p-1}}(1-{(-1)}^{a_1+E_1})(1-{(-1)}^{a_2+E_2})\cdots (1-{(-1)}^{a_m+E_m}).$$ We are interested in the mean value of the sums $$\sum _{c=1}^{p-1}{E}^2(c,m,p),$$ where $E(c,m,p)=N(c,m,p)-\left({(p-1)}^{m-1}\right)/\left(2^{m-1}\right)$ for the odd prime $p$ and any integers $m\ge 2$. When $m=2$, $c=1$, it is the Lehmer problem. In this paper, we generalize the Lehmer problem and use analytic method to give an interesting asymptotic formula of the generalized Lehmer problem.

Let $k$ be a finite extension of ${\mathbb{Q}}_p$, let $k_1$, respectively $k_3$, be the division fields of level $1$, respectively $3$, arising from a Lubin-Tate formal group over $k$, and let $\Gamma =$ Gal($k_3/k_1$). It is known that the valuation ring $k_3$ cannot be free over its associated order $𝔄$ in $K\Gamma$ unless $k={\mathbb{Q}}_p$. We determine explicitly under the hypothesis that the absolute ramification index of $k$ is sufficiently large.

We construct for a real algebraic variety (or more generally for a scheme essentially of finite type over a field of characteristic $0$) complexes of algebraically and $k$- algebraically constructible chains. We study their functoriality and compute their homologies for affine and projective spaces. Then we show that the lagrangian algebraically constructible cycles of the cotangent bundle are exactly the characteristic cycles of the algebraically constructible functions.