# Degree of a real algebraic variety and regular morphisms

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 $2max(d,e)$?

• Re 1: a point in $\mathbb{R}^3$ has degree 1 (with any reasonable definition) but it can't be given as the zero locus of a linear polynomial. – algori Nov 5 '11 at 17:05
• Off the top of my head, your definition does not look very nice. Let's say you have a variety which is intersection of two hypersurfaces of degrees $d$ and $e$. It would be natural to say that the degree of the intersection is $de$, which would be the degree of the complexification of your variety (at least when everything is nice). By your definition would have a degree of at most $2\max(d,e)$, so much smaller. I guess it depends what you want to use your degree for, but I don't see anything good coming out of this big discrepancy. – Thierry Zell Nov 5 '11 at 17:07
• 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 $2max(d,e)$? – probably Nov 5 '11 at 17:26
• @probably: absolutely not. First of all: your projection is a semi-algebraic set, not a variety in general. And the degrees you need will be higher in general. Projection is a pretty nasty operation. – Thierry Zell Nov 5 '11 at 17:31
• The Zariski closure of your semi-algebraic set (the projection) is indeed a real algebraic variety, which gives you also the complexfication. But again, this operation can be complicated. The reference for that is The complexification and degree of a semi-algebraic set by Marie-Françoise Roy and Nicolai Vorobjov. springerlink.com/content/g5u4rpfy0jp1dujm – Thierry Zell Nov 5 '11 at 21:54

## 1 Answer

I would encourage you to read Algorithms in Real Algebraic Geometry by Saugata Basu, Richard Pollack, Marie-Françoise Roy, which contains all the state of the art results about effective results in real algebraic geometry. It is a free download from http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html Projections are an instance of quantifier elimination, a procedure whose complexity is not entirely understood, but definitely very bad, even for a single existential quantifier.

What you will find is that things are a lot more tricky than you realize right now. In particular, there is a definite failure of Bézout-like theorems over the reals, and fact which clearly appears in Fulton's book.

Your definition dramatically underestimates the value of the degree. Here is an example derived from Fulton's book. Take $$f(x,y)= \prod_{i=1}^d (x-i)^2+\prod_{j=1}^d(y-j)^2.$$ Then, $V(f)$ has degree $\leq 2d$ by your definition, but it is made of $d^2$ isolated points. From a geometrc point of view, this is something whose degree should probably be at least $d^2$. The same example works in more variables, of course.