Multivariate sparse interpolation using randomized Kronecker substitutions

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by  Andrew Arnold

Multivariate sparse interpolation using
randomized Kronecker substitutions
4
1
0 Andrew Arnold Daniel S. Roche
2
Cheriton School of Computer Science Computer ScienceDepartment

y
University of Waterloo United States NavalAcademy
a
M Waterloo, Ontario, Canada Annapolis, Maryland, USA
[email protected] [email protected]

2
May 5, 2014

]
C
S
Abstract
.
s
c We present new techniques for reducing a multivariate sparse poly-
[ nomial to a univariate polynomial. The reduction works similarly to the

classical and widely-used Kronecker substitution, except that we choose
2
the degrees randomly based on the numberof nonzero terms in the mul-
v
4 tivariate polynomial. The resulting univariate polynomial often has a
9 significantly lower degreethantheKroneckersubstitutionpolynomial, at
6 the expense of a small number of term collisions. As an application, we
6 give a new algorithm for multivariate interpolation which uses these new
. techniquesalong with any existing univariate interpolation algorithm.
1
0
4
1 Introduction
1
:
v
We consider the problem of determining the coefficients and exponents of an
i
X unknown sparse multivariate polynomial, given a “black box” procedure that
r allows for its evaluation at any chosen point.
a
Our new technique is a variant on the classicalKronecker substitution. Say
f is an n-variate polynomial with max degree less than D and coefficients in a
ring R. The Kronecker substitution produces a univariate polynomial g R z
P r s
by substituting powers of z in the evaluation of f:
g z f z,zD,zD2,...,zDn´1 .
p q“
´ ¯
This map is invertible because there is a one-to-one correspondence between
termsinf anding,butthepriceofsuchconvenienceisanexponentialincrease
in the degree.
For example, consider the following bivariate polynomial:
9 5 4 6 2 9
f 3x y 2x y y x y .
“ ´ ´ `
1
The standard Kronecker substitution would be
10 19 45 60 92
g f z,z 3z 2z z z .
“ p q“ ´ ´ `
Every term in g comes from a single term in f, and the original exponents can
be determined by the base-D expansion of exponents in g.
Our randomized Kronecker substitution is also a map from multivariate to
univariate polynomials obtained by evaluating at powers of a single indetermi-
nate. Wechoosenintegers s1,...,sn atrandomandperformthe substitution
p q
g z f zs1,...,zsn . When these integers are not too large (in particular,
ifpeqac“h sp Dn´1), tqhe degree of g will be less than in the usual Kronecker
i
ă
substitution.
The price of a decreased degree is that the map is no longer invertible, for
two reasons. First, we may have two or more distinct terms in f converge to
a single term in g. This is called a collision. The second difficulty is that the
original multivariate exponents cannot be determined directly from the terms
in a single substitution g. We will show how performing O n log#f such
p ` q
random substitutions can overcome both difficulties.
In the example above, choose s1 5 and s2 2, so that
“ “
5 2 12 28 29 47
g5,2 z f z ,z z z 2z 3z .
p q“ p q“´ ` ´ `
In this case #g #f, so there were no term collisions,eventhough their order
“
has changed. The advantage is that the degree is significantly less than that
from the usual Kronecker substitution.
Choosing instead s1 2 and s2 5, the result is
“ “
2 5 23 30 49
g2,5 z f z ,z 3z 3z z ,
p q“ p q“ ´ `
which again has a reduced degree, but in this case we have a collision: The
two terms 2x5y4 y6 in f collided to produce a single term 3z30 under this
´ ´ ´
substitution.
Nonetheless, for the two terms not involved in a collision, both images can
be used to recover the original terms in f. The two terms with coefficient 3 in
the substitutions are 3z47 and 3z23. This produces the linear system
5 2 u 47
,
2 5 v “ 23
„ „  „ 
which is solved to reveal exponents u 9 and v 1 of the original term 3x9y.
“ “
Crucial to our success is determining a suitable set of integers from which
to choose the s . Section 3 provides bounds to construct such sets with prov-
i
ably many “good” choices which will not produce many collisions. Recovering
the original terms requires a way to correlate like terms in the images g, and
Section4showshowaseparaterandomizationallowsthecoefficientstobe used
to identify like terms in separate images g. Section 5 describes a multivariate
interpolation algorithm that makes use of these randomizations.
2
2 Related work
Polynomial interpolation dates from the 18th century, where the goal was to
discovera polynomialapproximationto anunknown function by collecting suf-
ficiently many observations. It is often beneficial to search for a sparse model,
whereonlyaboundednumberoftermsinthepolynomialarenonzero(see,e.g.,
Cand´es, Romberg, and Tao [2006]).
Our setting is more restrictive in two senses. First, we require the ability
to choose the points at which the unknown polynomial is evaluated. In this
black-box interpolationsetting,thepolynomialaprocedurewhichcanbeprobed
withanydesiredinput,atsomecost(see,e.g.,Grigoriev, Karpinski, and Singer
[1990], Mansour [1995], Giesbrecht, Labahn, and Lee [2009]).
We also require that a sparse polynomial truly exists “inside the box”, and
all evaluations — even those with numerical noise — come from the same T-
sparse polynomial. This is different from the general setting above where the
true function need not be T-sparseor indeed a polynomial at all. (Allowing for
outliers as in Comer, Kaltofen, and Pernet [2012] provides another option.)
Algorithmic progress in sparse interpolation can be divided into two cate-
gories depending on the complexity. Algorithms with polynomial dependence
on the partial degree bounds, sparsity bound, and number of variables are
called sparse algorithms [Zippel, 1979, 1990, Huang and Rao, 1999]. Those
with only logarithmic dependence on the degree are supersparse and can be
useful even in the case of univariate polynomials Ben-Or and Tiwari [1988],
Avendan˜o, Krick, and Pacetti [2006].
As shown in the introduction, Kronecker subtitution [Kronecker, 1882] can
turnaunivariate interpolationintoa multivariateone, byevaluating the multi-
variatepolynomialathighpowersoftheunivariateinterpolationpoints. Thisis
especiallyeffective inconjunctionwith supersparsealgorithms[Kaltofen, 2010].
Zippel’s sparse interpolation method proceeds one variable at a time and
usesrandomizationtoidentifywhichportionsoftheunknownsparsepolynomial
vanish. Thisisanotherwaytoturnanyunivariatealgorithmintoamultivariate
one Kaltofen and Lee [2003].
ThereisrecentworkonimplementationsofsparseinterpolationJavadi and Monagan
[2010], van der Hoeven and Lecerf [2014] and applications to GCD and factor-
ization [Javadi and Monagan, 2009, Berthomieu and Lecerf, 2012]. The Kro-
necker substitution in particular has been applied to integer and polynomial
multiplication [Scho¨nhage, 1982, Harvey, 2009].
Sparse interpolation has a strong connection to the important theoretical
problemofpolynomialidentity testing (see Shpilka and Yehudayoff[2010]for a
survey). Of particular interest is Klivans and Spielman [2001], where a similar
technique to ours is used for identity testing and interpolation. A significant
advantageof their algorithms is the very smallnumber of randombits required
for the computation.
Our randomized Kronecker substitution provides yet another way to trans-
form any univariate interpolation algorithm into a multivariate one. Table 1
summarizesour contributionas comparedto Klivans and Spielman[2001], Zip-
3
pel’smethod,andthenormalKroneckersubstitution. Thetableshows,foreach
method, how many times a univariate interpolation algorithm must be called,
andthedegreeoftheunivariatepolynomialsthatmustbeinterpolated,relative
to a given bound D on the max degree of the unknown multivariate polyno-
mial, and the number of variables n. (The sparsity bound for the univariate
algorithms will always be simply T.)
Table 1: Uni-to-multivariate interpolation methods
# of reductions Degree
Kronecker substitution 1 Dn
Zippel nT D
Klivans & Spielman n O n2T2D
p q
This paper (n 2 case) O logT O ?TDlogD
“ p q
This paper (n 3 case) O n logT O TD
ě p ` q ` p q ˘
The total cost of a multivariate interpolation algorithm obtained this way
depends on the choice of univariate interpolation algorithm. Writing D1 for
the univariate degree (the second column of Table 1), there are two options:
dense methods which require D1 black box probes and O D1 field operations,
p q
or supersparse methods based on Ben-Or and Tiwari which require only O T
2 p q
probes and O T log D1 field operations. In Table 2rwe compare two stan-
dard approaches against our own: Kronecker substitution with a supersparse
` ˘
univariate algrorithm, and Zippel’s method with a dense univariate algorithm.
Table 2: Overall multivariate interpolation
reduction univariate alg. # of probes # of field ops
Kronecker supersparse 2T O n2T log2D
2
Ours supersparse O nT O nT log D
p q ` ˘
r
Zippel dense nTD `O nTD ˘
r r p q
Ours (n 2 dense O ?TD O ?TD
“ q
Ours (n 3 dense O nTD Or nTD
ě q `p q˘ `p q˘
r r
r r
3 Randomized substitutions
For an unknown polynomial f R x1,...,xn , our main technical contribution
is a way of choosing integers Ps1,.r..,sn Zs such that the substitution g
P “
f zs1,...,zsn results in a lowerdegree than the usualKroneckersubstitution,
p q
while probably not introducing too many term collisions.
4
3.1 Bivariate substitutions
We beginwiththe caseofn 2 variables,whereourresultis strongerthanthe
“
general case and we always choose the random substitution exponents s1,s2 to
be prime numbers. Bivariate polynomials naturally constitute a large portion
of multivariate polynomials of interest, and they correspond to the important
case of converting between polynomials in Z x and multiple-precision integers
r s
[Scho¨nhage, 1982, Harvey, 2009].
Throughout this subsection, we assume f R x,y is an unknown bivariate
P r s
polynomial, written as
f a1xu1yv1 a2xu2yv2 atxutyvt. (3.1)
“ ` `¨¨¨`
WefurtherassumeupperboundsD ,D onthedeg f anddeg f,respectively,
x y x y
and T t on the number of nonzero terms #f.
ě
The general idea here is to perform the substitution
g z f zp,zq (3.2)
p q“ p q
forrandomchosenprime numbers p andq. We wantto choosep andq assmall
as possible, so as to minimize degg, but large enough so that there are not too
many collisions.
Ourapproachto choosingprimes is basedonthe followingtechnicallemma,
which shows how to guarantee a high probability success while minimizing the
degree of g.
Lemma 3.1. Let f R x,y with partial degrees less than D ,D and at most
x y
P r s
than T nonzeroterms, 0 µ 1be achosen boundon theprobability offailure,
ă ă
and 1 i T be the index of a nonzero term in f. Define
ď ď
25 T 1 lnD lnD
B p ´ q x y,
“ 9µ
BD BD
λ max 20.5, y and λ max 20.5, x
p q
“ ˆ c Dx ˙ “ ˜ d Dy ¸
By choosing primes p,q uniformly at random from the ranges λ ,2λ and
p p
r s
λ ,2λ respectively, the probability that the ith term of f collides in f zp,zq
q q
r s p q
is less than µ.
Proof. The primes p and q constitute the random choices in this discussion.
We say a pair p,q is bad if the term aixuiyvi collides with any other term in
p q
f zp,zq . We will show that the number of bad pairs p,q is at most µ times
p q p q
the total number of prime pairs that could be chosen.
We begin with a simple lower bound on the latter quantity. Applying equa-
tion(3.8)in Rosser and Schoenfeld[1962] twice showsthat the totalnumber of
ordered pairs p,q is at least
p q
9λ λ
p q . (3.3)
25lnλ lnλ
p q
5
Nextweobtainanupper boundonthe numberofbadpairsby countingthe
total number of times the ith term collides in every possible f zp,zq .
p q
ObservethatforanysetS ofnonzerointegers,andanyboundλ,thenumber
of times any prime p λ divides any of the integers in S is at most
ě
S lnmax S
| | p q. (3.4)
lnλ
Acollisionwiththeithtermoccurswheneveru p v q u p v q forsome
i i j j
` “ `
j i, 1 j T. This happens only when p v v and q u u .
i j i j
‰ ď ď (cid:23) p ´ q (cid:23) p ´ q
Since i j, these exponent differences u u and v v cannot both
i j i j
‰ p ´ q p ´ q
be zero. Furthermore, if one exponent difference is zero, then the collision can
never occur.
Therefore all collisions occur at indices in the set
J j 1 j T and u u v v 0 .
i j i j
“t | ď ď p ´ qp ´ q‰ u
The total number of times the ith term collides in any f zp,zq is equal to the
p q
sum over all j J of the number of pairs p,q such that p v v and
i j
P p q (cid:23) p ´ q
q u u .
i j
(cid:23)p ´ q
Now define, for each p, the subset of possible collision indices as J j
p
“ t P
J p divides v v .
i j
| p ´ qu
As each v v D , we have from (3.4) that
i j y
p ´ qă
#J T 1 lnD lnλ .
p y p
ďp ´ q {
pÿěλp
For each prime p, the total number of times the ith term collides is the
number of indices j J such that q u u . As each of these differences is
p i j
P (cid:23)p ´ q
less than D , using (3.4) again this sum is at most #J lnD lnλ .
x p x q
p ¨ q{p q
Therefore the total number of times the ith term collides is at most
#J lnD T 1 lnD lnD
p x p ´ q x y.
lnλ ď lnλ lnλ
q p q
pÿěλp
UsingthedefinitionofB andtheobservationthatλ λ B,wecanrewrite
p q
“
this bound as
9µ 9λ λ
B µ p q ,
25lnλ lnλ “ 25lnλ lnλ
p q p q
which is exactly µ times (3.3). Hence the probability of choosing a bad pair
p,q is at most µ.
p q
Choosing primes p,q fromsuch sets providesa goodbound onthe degree of
the resulting polynomial g.
Corollary 3.2. Let f R x,y with partial degrees less than D ,D and at
x y
P r s
most T nonzero terms.
6
Then for any constant error probability 0 µ 1, and primes p,q chosen
ă ă
randomly as in Lemma 3.1, the substitution polynomial g z f zp,zq has
p q “ p q
degree at most
O ?T D D log D D .
x y x y
p p qq
The degree of a standard Karonecker substitution is D D . Because T is
x y
always less than this, the randomized substitution will never have degree more
than a logarithmicfactor largerthan D D . The benefit comes when the poly-
x y
nomialissparse,i.e., T D D ,inwhichcasetherandomizedsubstitutionhas
x y
!
much smaller degree, albeit at the expense of a few collisions.
3.2 Multivariate substitutions
When f has at least 3 variables, the preceding analysis no longer applies. The
newdifficultyisthatpotentially-collidingtermscouldhaveexponentsthatdiffer
in two or more variables,meaning that the simple divisibility conditions areno
longersufficienttoidentifyeverypossiblecollision. Consequently,ourrandomly-
chosenexponentsinthiscasewillbesomewhatlarger,andnotnecessarilyprime.
For this subsection, f R x1,...,xn is an unknown n-variate polynomial,
P r s
written as
f a1xe1 a2xe2 atxet, (3.5)
“ ` `¨¨¨`
where x x1,...,xn and each ei Zn. D and T are upper bounds on the
“ p q P
max degree and number of terms in f.
Our general approach here is to choose a random vector s s1,s2,...,sn
“p q
of integers below a certain bound, and then perform the substitution g z
p q “
f zs1,...,zsn .
p q
The followinglemma,similarinpurposetoLemma3.1,showshowlargethe
integers in s must be in order to guarantee a small likelihood of collisions.
Lemma 3.3. Let f R x1,...,xn with max degree less than D and at most
P r s
T nonzero terms, 0 µ 1 be a chosen bound on the failure probability, and
ă ă
1 i T be the index of a nonzero term in f. Define λ to be the least prime
ď ď
number satisfying λ T µ.
ě {
If integers s1,...,sn are chosen uniformly at random from 0,λ 1 , then
r ´ s
the probability that the ith term of f collides in f zs1,...,zsn is less than µ.
p q
Proof. Adopt the notationF for the finite field with λ elements,which we will
λ
representasZ{λZ. Writes“ps1,...,snqfortherandomly-chosenvectorinFnλ.
Now let 1 j T, j i, and consider the jth term of f. Writing e ,e
i j
ď ď ‰
for the exponent vectors of these terms as in (3.5), define d e e , which
j i j
“ ´
cannot be the zero vector as i and j are distinct terms. We see that these two
terms collide in the substitution f zs1,...,zsn if and only if dj s 0.
p q ¨ “
Define ℓ 0 such that λℓ is the largest power of λ that divides every entry
in d , and wěrite d1 d λℓ. This means that d1 Zn and d1 modλ 0.
j j “ j{ j P j ‰
Furthermore, d s 0 if and only if d1 s 0.
j ¨ “ j ¨ “
Now,ifd1 s 0,thenthis alsoholdsmoduloλ, sosmustlieinthe n 1 -
dimension nujl¨l sp“ace of d1 modλ, call it W, where W Fn. The probpab´ilitqy
j Ă λ
7
that s W is 1 λ, and therefore the probability that terms i and j collide is at
P {
most 1 λ as well.
{
Since there are T 1 terms that the ith term could collide with, i.e., T 1
´ ´
choices for j, the probability that the ith term collides with any other term is
at most T 1 λ. From the definition of λ, this is less than µ.
p ´ q{
DuetoBertrand’spostulate,wewillhaveλ 2T µ. Thefollowingcorollary
ă {
shows how this bound on the size of entries in the randomly-chosen s affects
the degree of the reduced univariate polynomial. Compared to Dn, the degree
of the univariate polynomial from the usual Kronecker substitution, we see a
significant reduction when T Dn´1.
!
Corollary 3.4. Let f R x1,...,xn with max degree less than D and at most
P r s
T nonzero terms.
For any constant error probability 0 µ 1, and integers s1,...,sn chosen
ă ă
randomlyasinLemma3.3,thepolynomialg z f zs1,zs2,...,zsn hasdegree
p q“ p q
at most O TD .
p q
4 Multivariate diversification
Consider f R x1,...,xn as in (3.5):
P r s
f a1xe1 a2xe2 atxet.
“ ` `¨¨¨`
Each choice of s s1,...,sn for a randomized Kronecker substitution
“ p q
maps f to a univariate polynomial g z f zs1,...,zsn . In order to recover
p q “ p q
the exponent tuples of the original polynomial f, it will be necessary in the
next section to perform multiple such substitutions and correlateterms in each
g that correspond to the same unknown term in f.
The notion of diversification, introduced by Giesbrecht and Roche [2011],
will be used to correlate terms in the substituted polynomials g. The basic
idea is that distinct terms in f will be made, througha randomization,to have
distinct coefficients.
In fact, as there will be some small number of collisions in each substituted
polynomialg,werequirethenotionofgeneralizeddiversity fromArnold, Giesbrecht, and Roche
[2014]. The idea is that not only must the terms in f have distinct coefficients,
but some small number of sums of terms in f must additionally be distinct.
This problem of diversification is to choose α from a suitable set so that,
with high probability, α is not a root of any in a set H of polynomials. In
the original notion of diversity, H simply consists of the set of TpT´1q pairwise
2
termdifferencesfromf. To achievegeneralizeddiversity,wemustalsoconsider
polynomials h of the form
e e
h a x i a x j,
i j
“ ´
iPS jPS1
ÿ ÿ
where S and S1 each either comprise a single term or a set of terms appearing
in a collision.
8
Definition 4.1 (Diversifying set). Let n 1 and bounds D, m, and µ be
given, and let H R x1,...,xn be an unkněown subset of nonzero polynomials
satisfying #H Ďm arnd with eacsh h H having max degree less than D.
We say A ďR is a n,D,m,µ -dPiversifyingset if the probability is less than
Ď p q
µ that any evaluation point α, with entries chosen at random from A, is a root
of any of the h H. That is,
P
Pr h α 0 h H 1 µ.
αPAnr p q‰ @ P sě ´
From the discussion above, we see that the set H the differences between
anyofthet T singletermsandanyofther setsofcollisions,whichisatmost
#H 1 T ďr 2.
2
ă p ` q
Lemma 4.2. Let f R x1,...,xn with degree less than D and at most T
P r s
nonzero terms, and 0 µ 1. If there are at most r collisions in some set of
evaluations of f, and ăα ăRn is chosen at random from a set A R that is a
n,D,1 T r 2,µ -diverPsifying set, then, with probability at leastĎ1 µ, every
2
p p ` q q ´
coefficient off α1x1,...,αnxn is distinctfrom everyother coefficient andfrom
p q
the coefficients of all r collisions.
Proof. Follows from the discussion above and the definition of a diversifying
set.
Fromthe definitionofdiversifyingset, asimple applicationofthe Schwartz-
ZippellemmacouldbeusedtogeneratediversifyingsetsaslongasthefieldRis
sufficiently large. Theorems 3.1 and 4.6 in Giesbrecht and Roche [2011] define
diversifying sets for large finite fields and fixed-precision complex numbers, re-
spectively, in the univariate case n 1. Our more recent work in Arnold et al.
“
[2014] constructs smaller diversifying sets by choosing vectors of substitutions,
again for the univariate case. We restate these results in our current notation
and refer the reader to the aforementioned results for further details.
Corollary 4.3 (Giesbrecht and Roche[2011],Theorem3.1). Let bounds D, m,
and µ be given. If q is a prime power satisfying q mD µ, then the set F˚ of
ě { q
all nonzero elements in the finite field of size q is a 1,D,m,µ -diversifying set.
p q
Giesbrecht and Roche[2011]alsoconsidersthecasewhenf C x ,f isgiven
P r s
by a numerical black box. Their proof does not apply here as the polynomials
in H for us are not alwaysbinomials. We hope that a similar result would hold
for multivariate diversification, but do not consider the question here.
Corollary 4.4 (Arnold et al. [2014], Lemma 4.1). Let bounds D, m, and µ be
given as above, q a prime power, and set
s log 2D 1 and k log 2 2logm .
“ qp ` q “ µ `
Q U
Then the set Fk oPf k-tuples froTm a size-s extension of the finite field with q
qs
elements is a 1,D,m,µ -diversifying set˚.
p q
˚Inthiscasewemaketheabuseofnotationthateachevaluationfpαxqisactuallyak-tuple
ofevaluations, andthecoefficients infpαxqareactuallyk-tuplesinFqs.
9
We show, more generally, that vectorization may be applied to any diversi-
fying set.
Lemma 4.5. Let n,D,m,µ be given and H an unknown set of polynomials as
in Definition 4.1. If A R is a n,D,m,µ -diversifying set, then Aℓ Rℓ
is a n,D,m,µℓ -diversifĂying set, wphere additqion and multiplication in RĂℓ are
p q
component-wise.
Proof. As A is a 1,D,m,µ -diversifying set, then by definition, a randomly
selected row vectopr α Anqsatisfies h α 0 for all h H with probability
P p q ‰ P
at least 1 µ. Suppose β α1,...,αℓ is chosen randomly from Aℓˆn, and
´ “ p q
notethath β h α1 ,...,h αℓ . Thustheprobabilitythath β 0isthe
p q“ p q p q p q“
probability that h α 0 for every i 1,...,ℓ, which is at most µℓ.
i
`p q“ ˘ “
Rather than rehash the univariate diversification procedures, we refer the
readertotheaforementionedresultsandprovidethefollowingconnectionwhich
shows that univariate diversifying sets, with success probability scaled by a
factor of n, become multivariate diversifying sets.
Theorem 4.6. Let R be an integral domain and n,D,m,µ be given and H an
unknown set of polynomials as in Definition 4.1. If A R is a 1,D,m,µ n -
Ă p { q
diversifying set, then A is also a n,D,m,µ -diversifying set.
p q
Proof. The proof is by induction on n. When n 1, the statement holds
“
trivially. So assume n 2 and also that any 1,d,m,µ n 1 -diversifying set
ě p {p ´ qq
is also a n 1,d,m,µ -diversifying set.
Weknpow´thatHisqasetofn-variatepolynomials,eachwithmaxdegreeless
thanD. Rewriteeachhasapolynomialinx1 withcoefficientsinR x2,...,xn ,
r s
and define lc h to be the leading coefficient of h in terms of x1. Each lc h is
p q p q
an n 1 -variatepolynomialwithmaxdegreelessthanD. Furthermore,every
p ´ q
lc h must be nonzero since h 0.
p Nqowdefine H1 lc h h‰H . This isasetof n 1 -variatepolynomials
“t p q| P u p ´ q
with degrees less than D. Therefore, by the induction hypothesis, a random
evaluation point α2,...,αn with elements chosen from A is a zero for any
polynomial in H1pwith probaqbility less than n 1 µ n.
Now consider the set H2 h x1,α2,.p..,´αnq { h H , whose leading
coefficients areallnonzerowith“prtobpability atleastq1| µPn u1 . H1 is a setof
´ {p ´ q
at most m univariate polynomials with degrees less than D. From the original
definition of A, choosing α1 at random from A makes every h α1,α2,...,αn
p q
nonzero with probability at least 1 µ n.
Therefore a randomly-chosenpo´int{α1,...,αn An is a root of any h H
only if α2...,αn is a root of some poplynomial inqPH1, or α1 is a root of soPme
polynompial in H2.q As the probability of each of these is less than n 1 µ n
p ´ q {
and µ n, respectively, the probability either occurs must be less than µ, as
{
required.
10