Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$. We consider the image ${𝒩}_1(D,X)$ of ${𝒩}_1\left(D\right)$ in ${𝒩}_1\left(X\right)$ under the natural push-forward of $1$-cycles. We show that ${\rho }_X-{\rho }_D\le codim{𝒩}_1(D,X)\le 8$. Moreover if $codim{𝒩}_1(D,X)\ge 3$, then either $X\cong S\times T$ where $S$ is a Del Pezzo surface, or $codim{𝒩}_1(D,X)=3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that ${\rho }_X-{\rho }_T=4$.

We will prove that the Pierce-Birkhoff Conjecture holds for non-singular two-dimensional affine real algebraic varieties over real closed fields, i.e., if $W$ is such a variety, then every piecewise polynomial function on $W$ can be written as suprema of infima of polynomial functions on $W$. More precisely, we will give a proof of the so-called Connectedness Conjecture for the coordinate rings of such varieties, which implies the Pierce-Birkhoff Conjecture.

Given integers $D\>0,n\>1,0\<r\<n$ and a constant $C\>0$, consider the space of $r$-tuples $\overrightarrow{P}=(P_1...P_r)$ of real polynomials in $n$ variables of degree $\le D$, whose coefficients are $\le C$ in absolute value, and satisfying $\det {\left(\frac{\partial P_i}{\partial x_i}\left(0\right)\right)}_{1\le i,j\le r}=1$. We study the family $\left\{f\right|V\}$ of algebraic functions, where $f$ is a polynomial, and $V=\left\{\right|x|\le \delta ,\overrightarrow{P}(x)=0\},\delta \>0$ being a constant depending only on $n,D,C$. The main result is a quantitative extension theorem for these functions which is uniform in $\overrightarrow{P}$. This is used to prove Bernstein-type inequalities which are again uniform with respect to $\overrightarrow{P}$.The proof is based on...