Introduction
In thi s article , we shall understand b y a ring a commutative rin g with identity . Whe n
a ring k an d indeterminates X
v
. . . , Xn ar e given, the polynomial ring k[Xx, . . . , X n] in
Xv . . . , Xn over k i s a well defined rin g consisting o f al l polynomials i n thes e X
t
wit h co-
efficients i n k. Additio n an d multiplication ar e define d a s usual. On e remar k t o b e adde d
here is the following :
If f(Xlf . . . , Xn) E k[Xv . . . , X n] an d if K i s an arbitrar y rin g containing k, the n
f(Xt, . . . , Xn) define s a mapping o f Kn = K x ••• x K (set-theoretica l tt-ple product) into
K suc h that eac h {bv . . . , b
n
) i n Kn i s mapped t o f(Jbv . . . , b
n
). Now , addition an d
multiplication i n th e polynomia l rin g are define d b y th e correspondin g operation s in K,
namely, the su m (or th e product ) o f tw o polynomials map s (b
v
. . . , b
n
) t o th e su m (or
the product , respectively ) o f the image s of (b
v
. . . , b
n
) unde r thes e polynomials .
As one ca n se e immediately i n view of the definitio n o f th e polynomia l rin g
k[Xv . . . , X
n
], propertie s o f k hav e a lot t o d o wit h th e propertie s o f k[X
v
. . . , Xn] ;
a few representativ e example s wil l be discusse d i n th e firs t hal f o f §1.
In th e las t half o f §1, we review result s closely relate d t o Noether' s normalizatio n
theorem.
On the othe r hand , although i t i s important t o observ e polynomia l ring s without an y
restriction o n th e coefficien t rin g k, i t is also true tha t th e mos t basi c is the cas e where k i s
a field. I n tha t cas e we feel tha t th e polynomia l rin g k[Xv . . . , Xn] ha s very simpl e struc -
ture. Bu t thi s does not mea n tha t question s ar e simpl e i f the y ar e relate d t o polynomia l
rings over a field. Ther e ar e many difficul t problem s whic h ma y b e du e t o simplicit y o f th e
structure o f polynomia l rings . Thu s we are goin g to pa y mos t attentio n t o th e cas e where k
is a field. Thi s case corresponds t o th e affin e w-spac e (i.e., ^-dimensional affin e space ) ove r
k. Again , we feel tha t affin e space s have simpl e structur e an d ar e more understandabl e tha n
projective spaces . Bu t geometr y o f affin e space s is more understandabl e vi a geometry o f
projective spaces . Som e topics relate d t o thi s fac t wil l b e discusse d i n §2 .
As an exampl e o f complexit y o f phenomen a relate d t o polynomia l ring s or to rationa l
varieties, we shall discus s birational correspondence s o f rule d surface s i n §3 . Th e discussio n
will have a lot t o d o with automorphis m group s o f polynomia l ring s and Cremona group s
which wil l be discusse d i n § 4 and §5 .
Then w e shall discus s group action s o n affin e rings ; introductory remark s i n §6 , com-
plete reducibilit y o f rationa l representation s i n §7 , some fact s o n ring s of invariant s i n §8 ,
and som e supplementar y remarks , especially thos e relate d t o orbits , in §9 .
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