Nonlinear Equations - UFRJ

[SEC. 2.4: GROUP ACTION AND NORMALIZATION 19
It cannot have a degree smaller than d, for otherwise there would
be e < d, α ∈ k and G 0 , . . . G e−1 ∈ R with
x e n + G e−1 (y)x e−1
n + · · · + G 0 (y) = αf(y, x n ).
To see this is impossible, just specialize y = 0.
3: Fix an arbitrary y in k n−1 and solve f(y 1 , · · · , y n−1 , x) =
x d + f 1 (y 1 , . . . , y n−1 , x).
4: this is just Corollary 2.3.
5: In case f is irreducible, the discriminant in item 4 is not uniformly
zero. Hence in this case, for x 1 , . . . , x n−1 generic (in a Zariskiopen
set), there are d possible distinct values of x n for f(x) = 0.
The result above gives us a pretty good description of of hypersurfaces
in special position. Geometrically, we may say that when
f is irreducible, a generic ‘vertical’ line intersect the hypersurface in
exactly d distinct points. Moreover, generic n-variate polynomials are
irreducible when n ≥ 2.
2.4 Group action and normalization
The special position hypothesis f(x) = x d n+(low order terms) is quite
restrictive, and can be removed by a change of coordinates.
Recall that a group G acts (‘on the left’) on a set S if there is
a function a : G × S → S such that a(gh, s) = a(g, a(h, s)) and
a(1, s) = s. This makes G into a subset of invertible mappings of S.
When S is a linear space, the linear group of S (denoted by GL(S))
is the group of invertible linear maps.
We consider changes of coordinates in linear space k n that are
elements of the group GL(k n ) of invertible linear transformations
of k n . This action induces a left-action on k[x 1 , . . . , x n ], so that
(f ◦ L −1 )(L(x)) = f(x). If L ∈ GL(k n ), we summarize those actions
as
x L a(L, x) def
= L(x) and f L f ◦ L −1 .
This action extends to ideals and quotient rings,
J L J L def
= {f ◦ L −1 : f ∈ J}