I'm reading Fulton's "Intersection theory", which i need for some applied needs. And i have two questions on general definition of degree used in Fulton.

1)Let us we have a real algebraic variety defined by a set of equations $f_1=0, f_2=0,\ldots ,f_n=0$ of degrees $d_1,\ldots, d_n$ respectively. Using a well-known real-algebraic-geometry trick we can think about this variety as a variety defined by one equation $\sum_if_i^2=0$ of degree $2\max_{i}d_i.$ Then, we can take the smallest degree of all single polynomials representing fixed real algebraic variety as a degree of a variety.

Will this definition of degree coincide with given in Fulton "Intersection theory" $\S$ 8.4?

2)Let X be a real affine algebraic variety in $R^n$ of degree $p$ and let $f\colon R^n \to R^{n-1}$ be a projection. Will $deg \overline{f(X)}\leq \deg X$ in the sense of Fulton's definition? In sense of my definition?

$\overline{f(X)}$ here is a closure of $f(X)$ in Zarissky topology, not a semialgebraic one.

UPDATE

Being more exact, I have intersection of two hypersurfaces of degrees $d$ and $e$. And i want to project that intersection onto $R^{n−1}$. That projection will be(if everything is nice) a hypersurface in $R^{n−1}$. Can i say that this hypersurface could be represented by a polynomial of degree at most $2max(d,e)$?

semi-algebraic set,nota variety in general. And the degrees you need will be higher in general. Projection is a pretty nasty operation. $\endgroup$The complexification and degree of a semi-algebraic setby Marie-Françoise Roy and Nicolai Vorobjov. springerlink.com/content/g5u4rpfy0jp1dujm $\endgroup$1more comment