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TARSKI SEIDENBERG1
Oddly enough Seidenberg replacex elimination'mEGRof quantifiers with elimination on Ig
sits u
non equalities semi it.gs htClRA
p of 33Tp g
ptg tip g sl
to Eto
f O or g o
fg so
f s O Ngs o
f4gEO
Remember Rolle's theorem
pix so degg eggs
patplans where P'soMy.pe
xIgOMTI
So e.g pCx has a root almost
if pEx has no roots
Remark of real roots canbebounded
Tg of monomials Khovanski
iu Descartes oaixha at
most th number of alternations of
signs in the welficients positive roots
Multiple roots are harder to see
If p has a root so does p2
but small perturbation pR 25790 spbd.io
might not t methods
was
Remark IR dHer from in
terms of instability of roots
This is a serious practical
problempgtgi
Moduli Coefficients
ysxs.az xa
pT
Junstable0
stable 0
Is the zero at xs 1 an experimental
error
signH O
for X O a.e.gg unlike
a stable case
PCN has a double roof if
ged p p 1n
god can be determined alg
by Euclidean algorithm
Resultant
eyed f gI
sFa b c Iffy Fa b
of lowgod
i a.tossosfff
iand deg a deg g s gand deg b s deg f p
Make a matrix
Deg p t DegLg Deg2 ptg
a b 3 aft log
and check if diet 0 or not
Exampleat bite
vs dx te
f I p so HelixSget Caliphate
dateconst
Ex if
GCD 1pm p't dSturm's theorempass 11k g
In case f x ha no multiple roots p
so all roots are between the roots of f'c
INFORMALLY look for whether
ftp.lflpid 20 or not
But then you'd need to find the
p papi whee typoD
to continue to do th next degree
This can be done algorithmically
elegantly
STURM'S THEOREM
Better approachStant P pi B AP P R etc
inm
asb of rent in a b is ya HH
Examplep x s x3 x
P K Bx 1
Bs 4gXPy I b
sire x3 3 2 1 Ex 1
viii i pl
iV
v 2 t t t t so
Idlea of proof look at what happens to this as you passa root of p x it all zeroes are simple
or a root of some Pip s
i Ille
F g µQuad
coinsplato
Haruargot Plat play
imma
P es TEE
ChangerthrongharootoliffinkPisan
Goin Pilat Pilat Pilat
rovi.org e.fifmsmerod H Pg Pilat
of Pi Piafhanged
if P i Pit
KEY PROPERTIES
So do the same for P with multiple
roots
P P B Pr GCD Pi RIsf gGED Has the same of alternations of signs
asp p
for Pyp Pyp lr
when sticking out any a not a rootof P
Sturm works even if P has multipleroots
SYLVESTER's THEOREM
varieties Generalize 44 following
method from plane curves
old
Use a resultant to
get a 1 variable equation forsolve with Sturm
WHAT CAN GO WRONGF
Gf.no atIr sing pointf R Re
THIS IS HIGHLY NON GENERIC
Annals of Math 1954