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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!
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