ETNA
Electronic Transactions on Numerical Analysis.
Volume 25, pp. 201-205, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
ON NORMS OF FACTORS OF MULTIVARIATE POLYNOMIALS ON CONVEX
BODIES
ANDRÁS KROÓ
Dedicated to Ed Saff on the occasion of his 60th birthday
Abstract. Estimation of norms of factors of polynomials is a widely investigated extremal problem with numerous applications in functional analysis, number theory, approximation theory. In this note we study the following
problem: let be a convex body in and consider a product of polynomials , where is arbitrary and is a
monic multivariate polynomial. The goal is to find an upper bound for the uniform norm of on provided that
such bound for is known.
Key words. multivariate polynomials, norms of factors, convex bodies
AMS subject classification. 41A17, 41A63
1. Introduction. Estimation of norms of factors of polynomials is a classical extremal
problem which has been widely investigated for various norms. This extremal problem frequently arises in number theory, functional analysis, approximation theory, and therefore it
has been studied by many experts in these fields. (For corresponding results and references
see [1].) Most of the known results are related to univariate polynomials. In the present
note we shall consider this question for multivariate polynomials on convex bodies. A typical
problem
for norms"!#of
%$ factors can be formulated as follows: given a polynomial ,
on with factorization
&
('
-/. ) 10324
)+*,
provide sharp upper bounds for the norm of its factor 052 of degree at most 6 . In other words
we want to estimate the norm of 072 provided that the norm of its product with a monic
polynomial is given. The above question in one variable can be resolved using either Remez
or Markov inequalities.
A. Solution by the Remez inequality. Set
<' =
!
;
. ) ?A> @ . )CB ?D> @FEHG
)+*,
8:9 8N9
L
Clearly, I , , and for any KJ AM
and 032 as above we have
>
= ?A@ '
0 2 E G
>
O
Here and in what follows IOP
G#G3G stands for the Lebesgue measure in Q , while RTS#S3S#R3U
indicates the uniform
YX norm on V . Then by the Remez inequality (see [1]) with a proper
absolute constant W
we have
'
=
Y[A\ 2 9^_] ?D@
R%0#2R%Z
E G
>
` Received July 11, 2005. Accepted for publication November 28, 2005. Recommended by I. Pritsker. Supported
8:9
by the OTKA Grant T049196. This paper was partially written during the author’s visit at the Center of Constructive
Approximation,
Vanderbilt University, Nashville, TN, USA.
Alfred Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary
(kroo@renyi.hu).
201
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202
A. KROÓ
'
Thus setting a6cbd yields R%0#2R3Zefg
6d G
L
>
B. Solution by the Markov inequality. Let .
point. Then by the Markov inequality [1] for any be such that 052 attains its norm at this
L , ;hNiYjk.l , _ ! . B , _nm
d2 d2
poq
o
o
0324
0324
r.n st. R%0A2 u R3Z
R3032R3Zv -/. 6 d R%032R3Z
(1.1)
? R%0#2R3Z G
'
(
,
,
_
yz. ) is a monic
Furthermore, since is an interval of length at least 2 and w4
x;
d
)+*,
polynomial we have by the well known Chebyshev theorem
= '
o ? x
o ? '| ,
' | ,
'
b
b~
} E
R{wR Z ]
6 bd G
6 d
L , is such that w4
attains its uniform norm
Finally, this and (1.1) yield that whenever €
,
on at this point we have
?
? ' '
'
? ~ 6 d fg
6 d G
R%0#2R%Z
032
€
w4
€p
'
@
The asymptotic estimate fg
6‚d obtained above for R%0 2 R Z in case of a fixed
is in
general the best possible. Indeed,
it
is
shown
in
[3]
that
for
every
there
exist
polynoa
ƒ
„
W
ˆ
L and at the same time
2 of
whenever mials 0 o
X degree 6 such that … ‡†p0 2 X
@
R%0 2 R Z
6d‰† with' a proper positive constant depending only on ƒ . However, for large
the estimate fg
6 d becomes inefficient
o‹ and it can be replaced (using another method) by
|'
fg
Š 2
with a suitable constant Š
(see [1]).
2. New Results. Our goal is to solve the problem discussed above for multivariate polynomials on convex bodies. Both Remez and Markov inequalities are well developed in the
multivariate setting but the approach using Markov inequality seems to be more suitable for
several variables.
O
Thus we consider a convex body V in Q and the space of real multivariate polynomials
O
O
in Q of total degree at most 6 denoted as usual by Π2 . As in the univariate case we shall
L
O
Œ 2 with “monic” polynomials. We shall call a polynomial
consider products of 0#2
 C
Ž
.^”P ”•L Œ 2 O
 ] |c;; | ’ ‘#“ 2
Ž
.P” G
 ] |T;; |  ‘ * 2
!
Furthermore, let –s
r.  denote the ball in Q O with center at . and radius  , and for the convex
body V set
 rV—xh˜‰™pš›  P–s
. !  :œKV ! . L Vž !
monic if the sum of absolute values of its leading coefficients is 1, i.e.,
Ÿ
Vt& ¡¢#›
Ÿ
P–s
.
!Ÿ
v£KV
!
. L VH G
As above RcS3S3S7R U is the uniform norm on V .
O be a convex body, ¦§072  ' , where 0#2 and
T HEOREM 2.1. Let V¤œ¥Q
@
polynomials of degree 6 and , respectively, and in addition  ' is monic. Then
(2.1)
? @ B R30324R U
'
= #¨"© Ÿ
'
Vt
 V— d E 6 d 1R R U G
 ' are
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MULTIVARIATE POLYNOMIALS ON CONVEX BODIES
203
Note that just as in the univariate case estimate (2.1) yields an upper bound for 072 of order
'
@
fg
6d @ . This bound is sharp, in general, for fixed which is small relative to 6 , but for
large the following statement provides a more efficient upper bound. In order to formulate
it we need to recall
inequality by Kneser (see [1]) stating that for any univariate
! !+ª«#¬ a classical
polynomials 0 
0  ­6 and any interval ® we have
R%0pR#¯R  R3¯
V 2 R%0  R3¯
where V-2 is the Kneser constant. The Kneser constant is sharp in the above inequality and
2
its exact value can be found in [1]. Roughly, it grows as ° G ° .
T HEOREM 2.2. Under conditions of Theorem 2.1, the following estimate holds
R3032R U
V-2 |'
=
? ©
 Vt
E
'
R‡TR U G
R EMARK. It should be noted that the estimate R3072R U ±fg
6d
not hold, in general, when V is not convex. For instance, set
'
of Theorem 2.1 does
!
² ! !
Vs²l‹›n
p€ L Q d "W €
W  .-„ G
L Π, constructed in [3] such that
Consider the polynomial 072
2
²'
ph"! L !3%$³!7
oYX ' ² '
0#2
W
032
rWP
6 d G
!
'
L Œyd |' . Note that € ' is monic. Then clearly
Set €p;€ 0 2 2
‹"!
oYX ' ² '
R‡R3U&´
R%0 2 R3U&´
6 d G
Since .-„
the fg
6d
'
bound fails here for the nonconvex set V ² .
3. Proofs. In order to verify the new results we shall need some lemmas. The first
lemma is related to the geometry of convex bodies.

O
L EMMA! 3.1. Let V be a convex body in Q and W¶µ
Vt . Then for any L V
}^> Ÿ

the set –s
¸·FV contains a ball of radius ƒ/;
Vt+J V— .
>
> that –s
rW !  rV—+ L V and K‹
Proof.
Without
loss
of
generality
we
may
assume
¹ W . Set
? €s­
J
‰ G We claim that
>
!
!
–s
€ ƒ xœ
r–s
izV— G
>

! ?
L
Assume first that V— . Then for any º –s
€ J we have
>
^
? »
? K¼½A¾ ¿! 
º
J B J
› 
rV— G
>
>
>
! ?
!
L –s
€ ! J ? we clearly have
Hence –s
€ J CœY–•
W  V—+xœKV G Moreover, for every º
>
>
n
? B -/º
J
? > G
>
>
! ?
!
Thus we
have the inclusion –•
€ J eœq–s
G Therefore relation (3.1) holds when also
>
>
 rV— .
ever l ? Ÿ
!
Assume
now that „  V— . Note that V— . For any º L –•
€ ƒ set
?
À ; ºyt€p+J G Then
>
À ? 
ƒJ
rV— G
>
(3.1)
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204
Therefore À
A. KROÓ
L –s
rW !  rV—‰vœYV . Moreover
=
À
B
? > E B ? > À
ºÁ€
? >
!À L
ˆ ?
? ?
V and Wtµ J
J
Vt‰
J . Thus by the convexity of V we
where L V . Hence> –s
€ ! ƒÂœÃ
> V . Moreover, as above –s
€ ! ƒÂœÄ–s
€ ! J ? …œ
º
obtain
that
!
>
–s
which verifies (3.1) in this case, as well. The proof of the lemma is now complete.
>
Our next lemma provides a Chebyshev-type estimate for the minimal norm of a monic
multivariate polynomial. Its proof is based on a similar result by Kellogg for homogeneous
polynomials and the univariate Chebyshev inequality. It was shown by Kellogg [2] (see also
X ”
@
[4]) that for every homogeneous polynomial of degree given by Å
C
Ž
”n
 ] |T;; |  ‘ * '
we have
X n­© '
!
Ž
R3ÅR3Æ ‘Ç ]
”
 ] |c;; | ’‘ * '
O ,
O
where È b denotes the unit sphere in Q .
L
O

Π' be a monic polynomial. Then for any a
„ W and . L Q O we
L EMMA 3.2. Let '
>
have
9‰Í o ? 'x| , © '
b
R  ' R3ÉËÊ ²5Ì
J G
(3.2)
>
L È O b , , Î L Ï "!33$ set
Proof. For any À
!
'
' ,
w ' Ήx  ' À Î B .ËhŠ ' Î B Š ' b , Î b B S3S#S B ŠNÐ
>
where
Š ' ;
>
and .^” ’s are the leading coefficients of 
'
!
Ž
.^” À ”
 ] |T;; | ’‘ * '
' . By the well-known Chebyshev theorem we have
' n ? ' , '
? ' , '
b R{w R Z
b R R3ÉËÊ ²5Ì 9‰Í G
Š
Thus using the Kellogg inequality mentioned above and recalling that  ' is monic
n ©
' ? ' , '
b R R%É&Ê ²7Ì 9‰Í G
Ž
.P”
J >
 ] |T;; |  ‘ * '
This obviously yields (3.2).
P!
Proof of Theorem 2.1. Let R‡TR%U‹
R%0 2 R%U 0 2 hŠ
inequality proved by Wilhelmsen [5] on convex bodies for any À
!
L V . By a Markov-type
L È O b , ,€ L V
p ? N
Š 6d !
 Vt
ÑgÒ
where ! denotes the derivative in direction À . By this inequality for arbitrary ±
„ W and
L
>
€ –s
‚i•V
>
?
po
no
Š:6d
0#24
€p
Ša 0324
c/0324
€
Ša 
(3.3)
G
rV— >
ÑgÒ
0 2 €p
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205
MULTIVARIATE POLYNOMIALS ON CONVEX BODIES
}^Ÿ
 Vt+J V— such that
3.1 there exists a ball – of radius ƒ­;
Now according to Lemma
!
>
– is a subset of –s
4isV . Applying Lemma 3.2 for the monic polynomial  ' and ball –
>
we obtain
o ? 'x| ,
© '
b
R' RÉ
ƒJ G
L – . Thus choosing € L – so that the norm of
In addition, relation (3.3) holds for every €
 ' on – is attained at € we obtain by (3.3) and (3.4)
=
?
yoq
o
© '
6d ? 'x| ,

€p 0 2 €p ' €p
Š
 > E b
ƒJ G
rV—
Í
'xÓ Ê U Í _
Setting now 'x| , 2 and recalling the value of ƒ yields the needed estimate for ŠÔ
>
dÊ
R%0#2R U G
! L
V be such that
Proof of Theorem 2.2. Let €
¿!
R%0 2 R%U‹ 0 2 R  ' R%Uq  ' €p G
! $
By the convexity of V we have ®±;Õ € œÖV . Hence applying the Kneser inequality
mentioned above on the interval ® (and for corresponding univariate polynomials) we have
(3.5)
R%0 2 R%UgR  ' R%U
V 2 |' R1R3U G
Since  ' is monic and V contains a ball of radius  Vt we obtain using (3.2) with  rV—
>
together with (3.5)
(3.4)
R%0 2 R%U
'
? ©
V 2 |‚' J  Vt‰ R1R%U G
This completes the proof of Theorem 2.2.
REFERENCES
[1] P. B ORWEIN AND T. E RD ÉLYI , Polynomials and Polynomial Inequalities, Springer, New York, 1995.
[2] O. D. K ELLOGG , On bounded polynomials in several variables, Math. Z., 27 (1927), pp. 55–64.
[3] A. K RO Ó AND J. S ZABADOS , Markov-Bernstein type inequalities for multivariate polynomials on sets with
cusps, J. Approx. Theory, 102 (2000), pp. 72–95.
[4] M. R EIMER , Constructive Theory of Multivariate Functions, Wissenschaftsverlag, Mannheim, Wien,
Zurich, 1990.
[5] D. R. W ILHELMSEN , A Markov inequality in several dimensions, J. Approx. Theory, 11 (1974), pp. 216–
220.