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