RafaelOliveira

PrincetonUniversity

Factors of Low Individual Degree Polynomials

Outline

1IntroductionandBackground

Introductionand

Background 2Main Ideas ofthis Work

Main Ideas ofthis Work 3Conclusion&Open

Questions

Conclusion&Open

Questions

Introduction &Background

Arithmetic Circuits and Factoring

1

FactoringinRealLife

Basicroutineinmanytasks:

Usedtocompute:• PrimaryDecompositionsofIdeals• GröbnerBases,etc.

FastdecodingofReedSolomonCodes

Canbedoneefficientlyin(randomized)polytime!

Intheory,interestedin:• Derandomization• Parallelcomplexity• Structureoffactors

ArithmeticCircuits

Modelcapturesournotionofalgebraiccomputation

Definition bypicture

++

x y x-1

×f =y2– x2Mainmeasures:

Size=#edges

Depth=lengthoflongestpathfrom

roottoleaf

Itisamajoropenquestionwhetherhasasuccinctrep.inthismodel.

Permn

Manyinterestingpolynomialshavesuccinctrep.inthismodel,suchas.

Detn(X),�k(x1, . . . , xn)

PolynomialFactorization

Problem:Givenacircuitfor,where

outputcircuitsfor

P (x)

P (x) = g1(x)g2(x) . . . gk(x)

g1(x), g2(x), . . . , gk(x)

• [LLL’82,Kal ’89]:ifiscomputedbyasmallcircuit,thensoarethefactors.MoreoverKaltofengivesarandomizedalgorithmtocomputefactors

P (x)g1(x), g2(x), . . . , gk(x)

• Fundamentalconsequences to:• Circuit Complexity&Pseudorandomness:[KI’04,DSY’09]• CodingTheory:[Sud ’97,GS’06]• GeometricComplexityTheory:[Mul’13]

WhatAboutDepth?

Structure:givenpolynomial incircuitclass,whichclassesefficientlycomputethefactorsof?

P (x) CC⇤ P (x)

[Kaltofen’89]:factorizationbehavesnicelyw.r.t.size.

Whataboutdepth?

Moregenerally:

• Ifhasasmalldepthcircuit,doitsfactorshavesmalldepthcircuits?

• Ifhasasmallformula,doitsfactorshavesmallformula?

P (x)

P (x)

GapofUnderstanding

Generaldepthreductions[AV’08,Koi’12,GKKS’13,Tav’13]givesubexponentialgap.

Canthisbeimproved?

IfisapolynomialwithmonomialsanddegreeP (x) s d

Kaltofen&depthreduction

Factorsofcomputedbyformulasofdepth and

size.

P (x)4

exp(

˜O(

pd))

Polynomialswithboundedind. deg.formaveryrichclass,whichgeneralizesmultilinearpolynomials.Wellstudied,worksof[Raz ’06,RSY’08,Raz ’09,SV’10,SV’11,KS’152,KCS’15,KCS’16].

WhyBoundIndividualDegrees?

BoundedIndividualDegree

Multilinear

Trivial

Littleisknown

Steptowardsunderstandinggeneralcase

ThisWork

Theorem: Ifisapolynomialwhich:• hasindividualdegreesboundedby,• iscomputedbyacircuit(formula)ofsize&depthThenanyfactorofiscomputedbyacircuit(formula)ofsize

&depth

P (x)r

s df(x) P (x)

d+ 5

Furthermore,resultprovidesarandomizedalgorithmforcomputingallfactorsofintimeP (x)

poly(nr, s)

poly(nr, s)

PriorWork

[DSY’09]: ifiscomputedbyacircuitofsize,depth

• isboundedby

Thenitsfactorsoftheform havecircuitsofdepthandsize

P (x, y)

degy(P )

y � g(x)

ExtendHardnessvs Randomnessapproach of[KI’04]toboundeddepthcircuits.

r

s d

poly(nr, s)d+ 3

[DSY ’09] noticed thatonlyfactorsoftheformareimportanttoextend[KI’04] toboundeddepth.

y � g(x)

2 MainIdeasofthisWork

Lifting

RootApproximation

Reversal

Outline

Lifting

Supposeinput is:

Where

Howdowefactorinthiscase?

Cantrytobuildthehomogeneouspartsofoneatatime.

µ1 = g1(0), µ2 = g2(0) and µ1 6= µ2

P (x, y) = (y � g1(x))(y � g2(x))

gi(x)

Lifting

Notethat:

Whichweknowhowtofactor.

Hence,foundtheconstanttermsoftheroots.

P (0, y) = (y � µ1)(y � µ2)

Howtofindthelineartermsoftheroots?

Lifting

Settingintheinputpolynomial:

Since,theconstanttermof

isnonzero,whereastheconstanttermof

is zero!Hence,lineartermofequalsthelineartermof,uptoaconstantfactor.

y = µ1

µ1 6= µ2

µ1 � g1(x)

µ1 � g2(x)

P (x, µ1) = (µ1 � g1(x))(µ1 � g2(x))

P (x, µ1)g1(x)

Lifting

Continuingthisway,wecanrecovertherootsand factortheinputpolynomial.

Hensel Lifting/NewtonIteration.Pervasiveinfactoringalgorithms,suchas

[Zas ’69,Kal ’89,DSY’09],andmanyothers.

[DSY’09]: ifiscomputedbyacircuitofsize,depth

• isboundedby

Thenitsfactorsoftheform havecircuitsofdepthandsize

P (x, y)

degy(P )

y � g(x)

r

s d

poly(nr, s)d+ 3

Lifting

Twomainissues

• Whatifisnotmonic in?Usereversaltoreducethenumberofvariables

yP (x, y)

• Whatifdoesnotfactorintolinearfactorsin?

Approximaterootsinalgebraicclosureofbylowdegreepolynomialsin.

P (x, y)y

F(x)F[x]

2 MainIdeasofthisWork

Lifting

RootApproximation

Reversal

Outline

ApproximationPolynomials

Supposeinput is:

Whichdoesnot factorintolinearfactors.Let

where

Is irreducibleanddoesnotdividetheotherfactor.

P (x, y) = f(x, y)Q(x, y)

f(x, y) = yk +k�1X

i=0

fi(x)yi

P (x, y) = yr +r�1X

i=0

Pi(x)yi

ApproximationPolynomials

f(x, y) =kY

i=1

(y � 'i(x))

P (x, y) =rY

i=1

(y � 'i(x))

Whereeachisa“function”onthevariables'i(x) x

Anypolynomialfactorscompletelyinthealgebraicclosureof!F(x)

ApproximationPolynomials

Sinceandshareroots,cantrytoapproximatetheserootsbypolynomialsofdegreesuchthat

gi,t(x)P (x, y) f(x, y) 'i(x)

t

f(x, gi,t(x))

only hastermsofdegreehigherthan .t

Definition:wesaythat

ifthepolynomialonlyhastermsofdegreehigherthan.

f(x)� g(x)t

f(x) =t g(x)

ApproximationPolynomials

Thisdefinitiongivesusatopology:- Twopolynomialsarecloseiftheyagreeonlowdegreeparts

- Can usethistopology toderiveanalogsofTaylorseriesforelementsof!(#).

Can“approximate”elementsof!(#) bypolynomials!

Definition:wesaythat

ifthepolynomialonlyhastermsofdegreehigherthan.

f(x)� g(x)t

f(x) =t g(x)

ApproximationPolynomials

Thenwecanprovethefollowing:

Ifwecanfind foreachrootofsuchthat

gi,t(x) f(x, y)'i(x)

f(x, gi,t(x)) =t 0

Lemma:thepolynomialsaresuchthat

f(x, y) =t

kY

i=1

(y � gi,t(x))

gi,t(x)

Canconvertapproximationstotherootsintoapproximationstothefactors!

ApproximationPolynomials

Lookingatourparameters:

Howdoweobtainthesepolynomials?gi,t(x)

Since eachisalsoarootof,canobtainfromvialifting!

'i(x) P (x, y)gi,t(x) P (x, y)

Withstandardtechniques,canrecoverfrom

f(x, y)kY

i=1

(y � gi,t(x))

f(x, y) =t

kY

i=1

(y � gi,t(x))

Depthsized+ 4 poly(nr, s)

Observation:forthegeneralcase,needtokeeptheproducttopfanin!

2 MainIdeasofthisWork

Lifting

RootApproximation

Reversal

Outline

SetUp

Supposeinput nowis:

Let

where

is irreducibleanddoesnotdividetheotherfactor.

P (x, y) = f(x, y)Q(x, y)

P (x, y) =rX

i=0

Pi(x)yi, P0(x)Pr(x) 6= 0

f(x, y) =kX

i=0

fi(x)yi

TheGamePlan

Reduce tothemonic case:

P (x, y) = Pr(x) · yr +

r�1X

i=0

Pi(x)

Pr(x)yi!

f(x, y) = fk(x) · yk +

k�1X

i=0

fi(x)

fk(x)yi!

1. Recoverfrombysomekindofinduction2. Recoverthepartofthatdependson

fk(x) Pr(x)f(x, y) y

Thereexistswith�(x, y)

• depthd+ 4• size T (s, n)

NaïveRecursion

Lethaveindividualdegrees,variablesandcomputedbycircuitofsizeanddepths

r nP (x, y)d

Letbesuchthat:T (s, n)

f(x, y) | P (x, y)

�(x, y) =t f(x, y)

• topfaninproductgate

NaïveRecursion

Ourrecurrencebecomes:

Aftersteps,ourrecursionwouldbecomet

Exponentialwhen!t ⇠ n

T (s, n) T (3rs, n� 1) + poly(nr, s)

T (s, n) T ((3r)ts, n� t) + ⌦(ntrs)

Recoverfromfk(x) Pr(x)Sizeofpartdependingony

DealingwithExp.Growth

Howdoweavoidexponentialgrowth?

P0(x) = P (x, 0)

Itishardtoget from,butitiseasytogetfrom

P (x, y)P (x, y)P0(x)Pr(x)

hassmallercircuitsizethan!P0(x) P (x, y)

Whatifwecouldmaketheleadingcoefficientof?P (x, y)

P0(x)

Reversal

Thereversalcanbeefficientlycomputedfromcircuitcomputingoriginalpolynomial.

Definitionbyexample:If

Thenitsreversal isdefinedas

P (x, y) = P5(x)y5 + P4(x)y

4 + P0(x)

P̃ (x, y) = P0(x)y5 + P4(x)y + P5(x)

RecursionwithReversal

Aftersteps,ourrecursionremains

Noexponentialgrowth!

Ifwetakethereversaltocomputethefactors,ourrecurrenceforbecomesT (s, n)

t SizeofpartdependingonyRecoverfromf0(x) P0(x)

T (s, n) T (s, n� 1) + poly(nr, 9r2s)

T (s, n) T (s, n� t) + poly(nr, 9r2s)

2 MainIdeasofthisWork

Lifting

RootApproximation

Reversal

Outline

Outline

P (x, y) = f(x, y)Q(x, y) P̃ (x, y) = f̃(x, y)Q̃(x, y)

SizebecomesDepthremains

9r2sd

Monic iny

f̃(x, y) =t f0(x) · g(x, y)

Monic iny

P̃ (x, y) =t P0(x) ·G(x, y)

SizeDepthTopgate:productgate

poly(s, nr)d+ 4

Outline

Eachapproximaterootofisalsoapprox.rootof

g(x, y)G(x, y)

g(x, y) =t

kY

i=1

(y � gi,t(x))

SizeDepthTopgate:additiongate

poly(s, nr)d+ 3

Byinduction,f0(x) =t h(x)SizeDepthTopgate:productgate

poly(s, nr)d+ 4

SizeDepthTopgate:productgate

poly(s, nr)d+ 4

Outline

f̃(x, y) =t h(x) · g(x, y)

f̃(x, y) computedbycircuitof

SizeDepthTopgate:additiongate

poly(s, nr)d+ 5

3 Conclusions andOpenProblems

ThisWork- Recap

Weshowed: Ifisapolynomialwithindividualdegreesboundedby,andhasasmalllow-depthcircuit(formula),thenanyfactorofiscomputedbyasmalllow-depth circuit(formula).

P (x)rf(x) P (x)

Furthermore,resultprovidesarandomizedalgorithmforcomputingallfactorsofintimeP (x) poly(nr, s)

GeneralFramework

In[SY’10],itisaskedwhetherfactorsoflowdepthcircuitshavepolysizecircuitsoflowdepth,withouttheboundeddegreerestriction.

Theorem: Ifisapolynomialcomputedbyalowdepthcircuit,andallitsapproximaterootsarecomputedbysmalllowdepthcircuits,thenanyfactor ofiscomputedbysmalllowdepthcircuits.

P (x, y)

P (x, y)

Corollary:Tosettleaboveconjecture,itisenoughtosolvequestionaboveforapproximateroots,insteadoffactorsoftheform.

Questionopenevenforfactorsoftheform y � g(x)

y � g(x)

OpenQuestions

• Reducethedepthboundsintheworkof[DSY’09]• Canweshowthatfactorsofsparsehavesmall

depth4circuits?

• Derandomizepolynomialfactorization,evenforboundedindividualdegreepolynomials.• Questionisopenevenforsparsepolynomials• WillrequirestrongerPITsthancurrent

techniques

• Removeexponentialdependenceonthedegreeforfactorsoftheformy � g(x)

Thankyou!