Here is the materials of Prof. Vakil ‘s course on Foundations of Algebraic Geometry 07-08 in Stanford University:
Category: Algebraic Geometry
Algebraic geometry, semi-algebraic geometry, sums of squares, algebraic varieties,… Real analytic geometry, o-minimal structures,…
Classical Lojasiewicz inequalities
Let be a real analytic function with . Let and be a compact subset in . Then the (classical) \L ojasiewicz inequality asserts that:
- There exist such that
\noindent Let be a real analytic function with and . The \L ojasiewicz gradient inequality asserts that:
- There exist and a neighbourhood of such that
As a consequence, in (1), the order of zero of an analytic function is finite, and if is close to then is close to the zero set of . However, if is not compact, the latter is not always true and the inequality (1) does not always hold. The inequality (2) is similar to (1), it is not true in the case of is non-compact.
Two inequalities (1) and (2) have some special cases. For example, in the inequality (1), if has only isolated zero, i.e. , this implies . Hence, we have
On the other hand, being different from (2), we have another inequality:
There are some relations between and in complex case and real cases…
Curve selection lemma
There are many versions of the curve selection lemma.
In o-minimal structures, we have to consider definable curves, definable functions, deinable sets,… The definition of definable sets,… we can find in many documents.
Let be frontier of , ie . We have:
Curve selection lemma: In the o-minimal structure . If , then there is a definable map such that and .
Puiseux series
In this note, we discuss about Puiseux series and its appearance when we solve the equation . We refer to the book “Algebraic curves” of R.Walker.
Puiseux series are fractional power series:
where .
Order of series: .
Theorem. is algebraically closed.
( – the fieldof fractional power series).
By the proof of this theorem, we can see that (an algebraic curve), we can solve (Puiseux series) such that .
Nullstellensatz in the real case
In alegebraic geometry, we have Hilbert Nullstellensatz and in real algebraic geometry we have Real Nullstelllensatz. There is a difference between these theorems.
Strong Hilbert Nullstellensatz: .
If polynomial is vanish on the set (in ) then has the following form:
, that is: với .
In the case of
Real Nullstellensatz: .
:
or .
Weak Nullstellensatz
In algebraic geometry, one of the fundamental theorems is Hilbert’s Nullstellensatz.
Weak version:
Weak Hilbert’s Nullstellensatz: If is algebraic closed field then the maximal ideals of are exactly of the form for some .
Some cases:
If , a system of polynomials equation have a root ’cause .
If , there exists an ideal such that .
Another version:
\begin{theorem} Ideal with is algebraic closed. Then implies .
\end{theorem}
Moreover, , if we have a system of polynomials equation , if this one have no root then there exist
s.t.
For example (Parrilo): Consider following polynomials over :
,
,
.
There exist:
,
,
s.t. .