Multihomogenous Nonnegative Polynomials and Sums of Squares

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by  Alperen Ergur

MULTIHOMOGENOUS NONNEGATIVE POLYNOMIALS AND SUMS OF
SQUARES
ALPEREN A. ERGU¨R
Abstract. Werefineandextendquantitativebounds,onthefractionofnonnegativepoly-
nomials that are sums of squares, to the multihomogenous case. In particular, we start
laying the foundations for bounds incorporating arbitrary Newton polytopes. A key new
ingredient is an isotropic measure introduced by zonal harmonics.
6
1
1. Introduction
0
2
Let R[x¯] := R[x ,...,x ] denote the ring of real n-variate polynomials and let P denote
1 n n,2k
pr thevector spaceofforms(i.ehomogenouspolynomials)ofdegree2k inR[x¯]. Aformp P
n,2k
A is called non-negative if p(x¯) 0 for every x¯ Rn. The set of non-negative fo∈rms in
≥ ∈
2 Pn,2k is closed under nonnegative linear combinations and thus forms a cone. We denote
2 cone of nonnegative forms of degree 2k by Pos . A fundamental problem in polynomial
n,2k

optimizationandreal algebraicgeometryistoefficiently certifynon-negativity forreal forms,
]
G i.e., membership in Pos .
n,2k
A If a real form can be written as a sum of squares of other real polynomials then it is
. evidently non-negative. Polynomials in P that can be represented as sums of squares of
h n,2k
t real polynomials form a cone that we denote by Sq . Clearly, Sq Pos . We are
a n,2k n,2k ⊆ n,2k
m then lead to the following natural question.
[
Question 1.1. For which pairs of (n,k) do we have Sq = Pos ?
n,2k n,2k
2
v
In his seminal 1888 paper [12] Hilbert showed that the answer to Question 1.1 is affirmative
3
2 exactly for (n,k) ( 2 2N) (N 2 ) 3,4 . Hilbert’s beatiful proof was not con-
∈ { }× ∪ × { } ∪ { }
6 structive: The first well known example of a non-negative form which is not sums of squares
4
is due to Motzkin from around 1967: x6 +x2x2(x2 +x2 3x2).
0 3 1 2 1 2 − 3
. Hilbert stated a variation of Question 1.1 in his famous list of problems for 20th century
1
0 mathematicians:
6
1 Hilbert’s 17th Problem. Do we have, for every n and k, that every p Pos is a sum
n,2k
: ∈
v of squares of rational functions?
i
X
Artin and Schrier solved Hilbert’s 17th Problem affirmatively around 1927 [1]. However there
r
a isno knownefficient andgeneral algorithmforfinding theasserted collection ofrationalfunc-
tions for a given input p. Despite the computational hardness of finding a representation
as a sum of squares of rational functions, obtaining a representation as a sum of squares
of polynomials (when possible) can be done efficiently via semidefinite programming (see,
e.g., [15]). This connection to semidefinite programming (which has been used quite success-
fully in electrical engineering and optimization) strongly motivates a classification of which
(n,k) have membership in Sq occuring with high probability, relative to some natural
n,2k
probability measure µ on Pos .
n,2k
1.1. From All or Nothing to Something in Between. Hilbert’s 17th problem is
essentially an algebraic problem. However methods from analysis have recently enabled
PartiallysupportedbyNSF-MCSgrantDMS-0915245,REUgrantDMS-1156589,CCF-140902andNSF-
CAREER grant DMS-1151711.
1
2 ALPEREN A. ERGU¨R
some advances. The first example of this perspective is Gregoriy Blekherman’s work: A con-
sequence of his paper [6] is a probability measure µ on Pos supported in an hyperplane,
n,2k
for which µ(Sq ) 0 as n , for any fixed k 2.
n,2k → → ∞ ≥
It is important to observe that for many problems of interest in algebraic geometry, forms
with a special structure (e.g., sparse polynomials) behave differently from generic forms of
degree 2k. Precious little is known about Hilbert’s 17th Problem in the setting of sparse
polynomials [21, 17, 8]. So let us first recall the notion of Newton polytope and then a
theorem of Reznick: For any p(x¯) = c xα with α = (α ,...,α ) and x¯α = xα1 xαn,
α∈Zn α 1 n 1 ··· n
the Newton polytope of p is the convex hull Newt(p) := Conv( α c = 0 ).
α
P { | 6 }
Theorem 1.1. [17, Thm. 1] If p = r g2 for some g ,...,g R[x¯] then Newt(g )
i=1 i 1 r ∈ i ⊆
1Newt(p) for all i.
2 P
Thistheoremenablesustorefinethecomparisonofconesofsums of squaresandnon-negative
polynomials to be more sensitive to monomial term structure.
Definition 1.2. For any polytope Q Rn with vertices in Zn, let N := #(Q Zn),
Q
c = (c α Q Zn), p (x¯) = ⊂c x¯α and then define ∩
α | ∈ ∩ c α∈Q∩Zn α
Pos := c RNQ p (x¯) 0 for every x Rn
Sq := cQ R{PNQ∈ p (x¯)|=c q≥(x¯)2 where New∈t(q )} 1Q
Q { ∈ | c i i i ⊆ 2 } ⋄
In our notation here, Blekherman’s paper [6P] focused on volumetric estimates for the cones
Pos and Sq , where Q is the scaled (n 1)-simplex
Qn,2k Qn,2k n,2k −
x¯ Rn n x = 2k , x ,...,x 0 .
{ ∈ | i=1 i 1 n ≥ }
P
In this context the following problem arises naturally:
Weighted Polytopal SOS Problem. Given a polytope Q and a probability measure µ on
Pos , estimate µ Sq .
Q Q ⋄
Note that Hilbert’s(cid:0)class(cid:1)ic work [12] implies that, for any n and k and any continuous nonnegative
probability measure on the cone Pos , we have µ Sq = µ Sq = µ Sq = 1.
n,2k Q3,4 Q2,2k Qn,2
A related variant of the Weighted Polytopal SOS P(cid:16)roblem(cid:17)was r(cid:16)ecently (cid:17)answe(cid:16)red, as(cid:17)a con-
sequence of the main theorem from [8]: There is now a complete combinatorial classification
of those polytopes Q for which Sq = Pos .
Q Q
1.2. Our Results. We focus on the multihomogenous case of the Weighted Polytopal SOS
Problem. We have leaned toward general methods rather than ad hoc methods, in order to
allow future study of arbitrary polytopes. We begin here with Cartesian products of scaled
standard simplices.
Definition 1.3. Assume henceforth that n = n + + n and k = k + + k , with
1 m 1 m
k ,n N for all i, and set N := (n ,...,n ) and·K·· := (k ,...,k ). We··w·ill partition
i i 1 m 1 m
∈
the vector x¯ = (x ,...,x ) into m sub-vectors x¯ ,...,x¯ so that x¯ consists of exactly n
1 n 1 m i i
variables for all i, and say that p R[x¯] is homogenous of type (N,K) if and only if p is
∈
homogenous of degree k with respect to x¯ for all i. Finally, let Q := Q Q .
i i N,K n1,k1 ×···× nm,km ⋄
Example 1.4. p(x¯) := x3x2+x x2x2+x3x x is homogenous of type (N,K) with N = (3,2) and
1 4 1 2 5 3 4 5
K = (2,3). (So x¯ = (x ,x ,x ) and x¯ = (x ,x ).) In particular, Newt(p) Q = Q Q .
1 1 2 3 2 4 5 N,K 3,2 2,3
⊆ × ⋄
MULTIHOMOGENOUS NONNEGATIVE POLYNOMIALS AND SUMS OF SQUARES 3
Multihomogenous forms appeared before in the work of Choi, Lam and Reznick. In par-
ticular they proved the following theorem in [7]:
Theorem 1.5. (Choi, Lam, Reznick) Let N = (n ,n ,...,n ) and K = (2k ,2k ,...,2k )
1 2 m 1 2 m
where n 2 and k 1 then Pos = Sq if and only if m = 2 and (N,K) is either
i ≥ i ≥ QN,K QN,K
(2,n ;2k ,2) or (n ,2;2,2k ).
2 1 1 2
Our result can be viewed as a localized version of Blekherman’s Theorem and also as a
quantitative version of the theorem of Choi, Lam and Reznick [7]. In order to state our
result we need to introduce the following function on subsets of P .
N,K
Definition 1.6. Let Sn−1 denote the standard unit (n 1)-sphere in Rn and define
S := Sn1−1 Snm−1, P := p R[x¯] homogeneous−of type (N,K) , and C :=
N,K N,K
×···× { ∈ }
1
p P pdσ = 1 . For any X P we set µ(X) = vol(X∩CN,K) DN,K where D
∈ N,K | S ⊆ N,K vol(B) N,K
(cid:8)is the dimensRion of PN,K(cid:9)and B is the DN,K dimensional bal(cid:16)l. ⋄ (cid:17)
Our main theorem is the following.
Theorem 1.7. Let N = (n ,n ,...,n ) and K = (2k ,2k ,...,2k ). We define L :=
1 2 m 1 2 m QN,K
p Pos : p = l2k1l2k2 l2km where l is a linear form in variables x¯ , then the
{ ∈ QN,K i i1 i2 ··· im ij j}
following bounds hold
P
m
1
4m√max n (2ki +1)−21 ≤ µ(PosQN,K) ≤ co
i i
i=1
Y
m m
n ck
c1 (2ki + 2i)−k2i ≤ µ(SqQN,K) ≤ c2 (n +ik )k2i
i i
i=1 i=1
Y Y
m m m
n n
c3 (2ki + 2i)−k2i ≤ µ(LQN,K) ≤ miaxni 4(2ki +1)12 (2kii)−k2i
i=1 i=1 i=1
Y q Y Y
where c are absolute constants with c 5 and c = 210e.
i o
≤
To compare our bounds with Blekherman’s bounds [6] let us consider the following special
cases of Theorem 1.7:
Corollary 1.8. (1) For N = (2,n 2) and K = (2k 2,2) we have the following bounds:
− −
µ Sq
−k+1 −1 QN,K
c1(2k −1) 2 (n+2) 2 ≤ µ(cid:16)Pos (cid:17) ≤ 1
QN,K
(2) Assume n = k.n , we partition into k group(cid:0)s by sett(cid:1)ing N = (n ,n ,...,n ) and
1 1 1 1
K = (2,2,...,2), then we have the following bounds:
µ Sq
n −k QN,K n −k+1
c1(2+ 2k) 2 ≤ µ(cid:16)Pos (cid:17) ≤ c2(ck) 2
QN,K
where c, c and c are absolute constants(cid:0)with c =(cid:1)210e.
1 2 48
4 ALPEREN A. ERGU¨R
Note that both cases considered above are contained in Q . In particular, Blekherman’s
n,2k
Theorems 4.1 and 6.1 from [6] give the following estimates:
nk+21 c1k!(k 1)! µ SqQn,2k c242k(2k)!√k −k+1
− n 2
(n +2k)k 42k(2k)! ≤ µ(cid:16)Pos (cid:17) ≤ k!
2 Qn,2k
where c1 and c2 are absolute constants. (cid:0) (cid:1)
The first case in Corollary 1.8, is a modest example to show the reflection of monomial
structure in our bounds. Bounds in the second case is dependent on n instead of n which
k
shows the effect of underlying multihomogeneity. In particular in the cases that k and n are
comparable bounds behave significantly different then the bounds of Blekherman.
In general, Theorem 1.7 proves that if we assume multihomogeneity on the set of variables
x¯ ,x¯ ,...,x¯ , bounds derived in Blekherman’s work for the ratio of sums of squares to
1 2 m
non-negative polynomials, holds locally for every set of variable x¯ .
j
The remainder of this paper is structured as follows: In Section 2 we define two differ-
ent inner products, and investigate basic relations between geometries introduced by these
two inner products. Section 2 also includes definiton of zonal harmonics and their basic
properties. The hurried reader can see the definitions at the very beginning and then go to
Lemmata 2.8, 2.11, 2.13 and 2.14. In Section 3 we prove volumetric bounds for Pos .
QN,K
A key step is discovering existence of an isotropic measure linked to the zonal harmonics.
In Section 4 we give bounds for Sq via classical convex geometry. Section 5 is devoted
QN,2K
to polynomials that are powers of linear forms. The bounds there are derived by a simple
duality observation.
2. Harmonic Polynomials and Euclidean Balls
In this section we develop necessary background for the proof of Theorem 1.7. We are
going to make use of two different inner products on P , mainly due to two useful notions
N,K
of duality.
Definition 2.1. For f,g P the “usual” inner product is defined as f,g := fg dσ.
N,K
∈ h i
S
For f(x) = c xα P with α = (α ,...,α ) we define the linear differential Roperator
α α ∈ N,K 1 n
D[f] := c ∂α1 ∂αn , and set f,g := D[f](g). This way of defining f,g ,
αPα ∂xα11 ··· ∂xαnn h iD h iD
introduces an in(cid:16)ner product w(cid:17)hich we call the “differential” inner product.
P ⋄
If g(x) = b xα P and f(x) = c xα P then it is easily checked that
α α ∈ n,d α α ∈ n,d
c b
P P α α
f,g = d!
h iD d
α1+·X··+αn=d α1,...,αn
where d is the multinomial coefficient d! .(cid:0) (cid:1)
α1,...,αn α1!···αn!
Below we list some useful properties of the differential inner product. This inner product
(cid:0) (cid:1)
is actually the Bombieri-Weyl inner product (see, e.g., [19]) in our setting.
Lemma 2.2. For any vectors v¯,x¯ S we define
∈
K (x) =(v x + +v x )2k1(v x + +v x )2k2 (v x + +v x )2km.
v 1 1 ··· n1 n1 n1+1 n1+1 ··· n n1+n2 ··· n−nm+1 n−nm+1 ··· n n
Then
(1) p,K (x) = p(v)
v D
h i
MULTIHOMOGENOUS NONNEGATIVE POLYNOMIALS AND SUMS OF SQUARES 5
(2) pq,h = deg(q)! q,D[p](h)
h iD deg(h)!h iD
(3) Let s := i n and s := 0 let for any i ∆ = ∂2 + + ∂2 and r =
i j=1 j 0 i ∂x2si−1+1 ··· ∂x2si i
(x2si−1+1 +·P··+x2si)21 then (p+2)(p+1)hri2p,qiD = hp,∆i(q)iD
We would like to compare the geometry induced by the two different inner products. For
instance how are the Euclidean balls with respect to two different products related to each
other? To find out, we will need to introduce a generalization of harmonic polynomials that
applies to our multihomogenous setting, and define corresponding linear operators acting on
underlying vector spaces.
We call f Π-harmonic if ∆ (f) = ∆ (f) = = ∆ (f) = 0. One may suspect that
1 2 m
···
Π-harmonicity is too strong a condition to be as natural as ordinary harmonicity. However,
one observes that Π-harmonicity is a special case of Helgason’s general theory of Harmonic
polynomials (see Chapter 3, [13]) .
Let H be the space of Π-harmonic polynomials in P . We observe that H is a
N,K N,K N,K
vector space by the linearity of the ∆ operators. Also, for any vector K = (k ,...,k )
i 1 m
(N 0 )m we define ∈
∪{ }
= Q (N 0 )m Q = (q ,...,q ), 2k q and q is even. .
K 1 m i i i
K { ∈ ∪{ } | ≥ }
Lemma 2.3. For U,Q such that Q = U we have that H and H are orthogonal
K N,U N,Q
∈ K 6
with respect to the usual inner product.
Proof. Let f H and g H with Q = U. Without loss of generality, assume
N,Q N,U
∈ ∈ 6
Q = (q ,...,q ) , U = (u ,...,u ) and q = u . Then we have
1 m 1 m 1 1
6
(q u ) fg dσ = (q fg u fg)dσ
1 1 1 1 1 1
− −
Z Z
Sn1−1 Sn1−1
(q fg u fg)dσ = (fD (g) gD (f))dσ = (f∆ g g∆ f)dσ = 0.
1 1 1 n n 1 1 1 1
− − −
Z Z Z
Sn1−1 Sn1−1 Bn1−1
So fgdσ = 0 and thus
1
Sn1−1
R
fg dσ = fg dσ dσ = 0
1 m
··· ···
Z Z Z
S Snm−1 Sn1−1
(cid:4)
Toseetheconnectionbetween ourtwo inner productsweuseavariant ofamapintroduced
by Reznick [18]: Let T : P P be defined via
N,K N,K
→
T(f)(x) = A−1 f(v)K(v,x) dσ(v)
Z
Sn1−1×...×Snm−1
where A = x2k1x2k2 x2km and K(v,x) as defined in Lemma 2.2.
n1 n1+n2··· n
Sn1−1×Sn2−1···×Snm−1
R
The following lemma shows that the operator T captures the relationship between our two
inner products.
6 ALPEREN A. ERGU¨R
Lemma 2.4. For all f,g P we have T(f),g = A−12k ! 2k ! f,g .
N,K D 1 m
∈ h i ··· h i
Proof. The case of arbitrary m is only notationally more difficult than the m=2, thanks to
induction. So we assume without loss of generality than m=2.
T(f),g = A−1 f(v)vK dσ,g = A−1 f(v)vK,g dσ
D D
h i h i
* +
Z Z
Sn1−1×Sn2−1 D Sn1−1×Sn2−1
Since vK,g = 2k !2k !g(v) we have
D 1 2
h i
2k !2k !
A−1 f(v)vK,g dσ = A−1 2k !2k !f(v)g(v)dσ = 1 2 f,g
D 1 2
h i A h i
Z Z
Sn1−1×Sn2−1 Sn1−1×Sn2−1
(cid:4)
The lemma above immediately tells us T is one-to-one since for f,h P , assuming
N,K
∈
T(f) = T(h) implies f,g = h,g for all g P . To prove our next lemma we need to
N,K
h i h i ∈
recall a theorem of Funk and Hecke.
Theorem 2.5. [20] Given any measurable function K on [ 1,1] such that the integral
−
1
K(t) (1 t2)n−22 dt
| | −
Z
−1
is well-defined, for every function H that is harmonic on Sn we have
1
K(σ ζ)H(ζ) dζ = Vol Sn−1 K(t)Pk,n(t)(1 t2)n−22 H(σ)
·  − 
Z Z
Sn (cid:0) (cid:1)−1
where P (t) is the classical Gegenbauer (ultraspherical) polynomial. 
k,n
Gegenbauer polynomials are naturally introduced by zonal harmonics and they do exist
more generally in spaces of sparse polynomials as well. Zonal harmonics and ultraspherical
polynomials in our setting are introduced in Lemmata 2.12, 2.13, 2.14.
Lemma 2.6. For f H we have
N,K
∈
T(f)(x) = E f(x)
N,K
where E depends only on N and K.
N,K
Proof. Defining K (v,x) = (v x + +v x )2ki, we have
i si−1+1 si−1+1 ··· si−1+ni si−1+ni
m
T(f)(x) = A−1 f(v) K (v,x) dσ (v)
i i
···
SnZm−1 SnZ1−1 Yi=1
We observe that each K satisfies the assumptions of the Funk-Hecke formula. So we have
i
m
T(f)(x) = A−1 ... A f (v,x) K (v,x) dσ (v)
1 1 i i
SnZm−1 SnZ2−1 Yi=2
MULTIHOMOGENOUS NONNEGATIVE POLYNOMIALS AND SUMS OF SQUARES 7
here f (v,x) means first n variables are fixed to x ,...,x and the rest left as variables
1 1 1 n1
1
in the integral and A1 = (cid:18)dvimol((HSnn11,−2k11))−1 t2k1Pn1,2k1(t)(1−t2)n12−2 dt(cid:19) where Pn1,2k1 is the
corresponding ultraspherical polynomialR. Iterating the argument we have
1
T(f)(x) = A−1 Aif(x) = A−1 dvimol((HSni−1)) t2kiPni,2ki(t)(1−t2)ni2−2 dtf(x)
i i ni,2ki Z
Y Y −1
(cid:4)  
Thanks to Lemma 2.6 we know that Π-harmonic polynomials are eigenvectors for T.
We also know a relation between our two inner products thanks to Lemma 2.4, and the
orthogonalityofspacesofΠ-harmonicpolynomialswithrespecttousualinnerproductthanks
to Lemma 2.3. We thus immediately obtain the following corollary.
Corollary 2.7. For U,Q such that Q = U we have that H and H are orthogonal
K N,U N,Q
∈ K 6
with respect to the differential inner product.
Remark 2.1. For notational convenience we set r = r r where r is as defined in
1 m i
Lemma 2.2 and let rα = rα1 rαm for any α Zm. ···
1 ··· m ∈
Forg SO(n ) SO(n ) and f P we define g f by setting g f(x) = f(g−1(x)).
1 m N,K
∈ ×···× ∈ ◦ ◦
This gives a well defined group action on vector spaces of polynomials. We observe that the
operator T commutes with SO(n ) SO(n ) action.
1 m
×···×
Thisimplies thatT(rK) = arK forsome constantasince the only polynomialsthat are fixed
under the action are constant multiples of rK. To compute a we check
1 = T(rK)(e + +e ) = arK(e + +e ) = a. Hence r is fixed under T. Lemma 2.4
n1 ··· n 1 ··· n
implies that the hyperplane orthogonal to r is fixed also.
⋄
Now we prove a decomposition lemma for P in a Hilbert space sense.
N,K
Lemma 2.8. P = rK−αH
N,K α∈KK N,α
Proof. First observe thLat rK−αH rK−βH for any α = β by Lemma 2.3. Also via
N,α N,β
⊥ 6
iterated usage of third property in Lemma 2.2 we observe rK−αH rK−βH . So direct
N,α D N,β
⊥
sum makes sense for both of the inner products. Let E = rK−αH and assume
α∈KK N,α
E = P . This implies there exists p P such that p E. By assumption p / H
N,K N,K D N,K
6 ∈ ⊥L ∈
therefore there exist i such that ∆ (p) = 0. Wlog say ∆ (p) = p = 0. Then if p is Π-
i 1 1 1
6 6
harmonic we have p,r2p = ∆ (p),p = p ,p = 0. This yields a contradiction
h 1 1iD h 1 1iD h 1 1iD 6
since r2p E. If p is not Π-harmonic there exist a j such that ∆ (p ) = 0. Wlog say
1 1 ∈ 1 j 1 6
∆ (p ) = p . Repeating the same argument, we arrive to a contradiction surely since all
1 1 2
(cid:4)
polynomials of degree 0 are Π-harmonic!
Corollary 2.9. P (S) = H (S) where P (S) is restriction of P to S =
N,K α∈Km N,α N,K N,K
Sn1−1 Sn2−1 Sn3−1 ... Snm−1.
× × × ×L
FromLemma 2.8 we know how P is decomposed into Π-harmonicpolynomials. We also
N,K
know T(f) = E f for any f H . For f rK−αH , since T is averaging over S where
N,K N,K n,α
∈ ∈
rK−α is constant, repeating Funk-Hecke argument in Lemma 2.6 we observe T(f) = C f
N,α
for some constant depending on N and α only. We would like to compute these constants
and write the operator T explicitly. Thankfully the integrals that gives the constants are
8 ALPEREN A. ERGU¨R
well known and computed. (See for example Lemma 7.4 of [5]). We write the result without
proof.
Lemma 2.10. Let f be the projection of f onto the subspace H then we have
α N,α
m k !Γ(ni+2ki)
T(f) = i 2 rK−αf .
(k αi)!Γ(ni+2ki+αi) α
αX∈KKYi=1 i − 2 2
Now let f = rK−αf that is f 2 = f,f = f 2. We set
α∈KK α k k h i α∈KK k αk
aα = mi=1 (ki−kα2ii!)Γ!PΓ(n(in+i2+2k2i2k)i+αi). Then define AN,K = maxα∈KK aα ,PBN,K = minα∈KK aα and
C = A−1 m 2k !.
N,K Q i=1 i
Q
A C f 2 T(f),T(f) = C f,T(f) B C f 2
N,K N,K D N,K N,K N,K
k k ≥ h i h i ≥ k k
We denote the ball with respect to the usual inner product by B and the ball with respect
to the differential inner product by B . The observation above implies
D
1 1
T(B) B T(B)
D
A C ⊆ ⊆ B C
N,K N,K N,K N,K
Lemma 2.11. Let T bepthe operator on P as depfined before Lemma 2.4, let A , B ,
N,K N,K N,K
C be defined as above , and let dim be the dimension of P , then we have the following
N,K N,K
A = 1
N,K
m ni +2k −1
B = 2 i
N,K
k
i=1(cid:18) i (cid:19)!
Y
m 1 k2i 1 m 1 k2i
det(T) dim
i=1(cid:18)2ki + n2i(cid:19) ≤ | | ≤ i=1 1+ 2nkii!
Y Y
m 1 k2i BD 1 k m 1 k2i
i=1(cid:18)2ki + n2i(cid:19) ≤ CN,K(||B||)dim ≤ e2 i=1 1+ 2nkii +1!
Y p Y
Proof.
m k !(ni +k )!
a = i 2 i
α (k αi)!(ni +k + αi)!
i=1 i − 2 2 i 2
Y
It is quite clear that a is maximized for α = (0,0,...,0) and minimized for α = K.
α
Thus AN,K = 1 and BN,K = mi=1( n2i+ki2ki )−1. Also T is a diagonal operator in the basis of
harmonic polynomials and it’s entries are a . Thus
Q (cid:0) (cid:1) α
T(B)
| | = det(T) = a dim(Hn,α)
α
B | |
(cid:12) (cid:12)
| | (cid:12)αY∈KK (cid:12)
(cid:12) (cid:12)
We observe K = (k1 +1)(k2 +1) (km +(cid:12)1) , (cid:12)
|K | ··· (cid:12) (cid:12)
MULTIHOMOGENOUS NONNEGATIVE POLYNOMIALS AND SUMS OF SQUARES 9
m n2i+2ki
a = ki−α2i
α (cid:0)n2i+2ki(cid:1)
Yi=1 ki
this yields the formula
(cid:0) (cid:1)
dim(HN,α)
det(T) di1m = B m n2i +2ki dim(PN,K)
| | N,KαY∈KK Yi=1(cid:18)ki − α2i (cid:19)!
If we partition into k +1 subsets by defining := α : α = 2k 2j then
K 1 j K 1 1
K K { ∈ K − }
we have
det(T) di1m = B ki a = B ki n2i +2ki ki1+1A
N,K α N,K
| | j
Yj=0αY∈Kj Yj=0(cid:18) (cid:19)
For some A determined by n and k for i 2. We repeat the same trick for A and do
i i
≥
some housekeeping to arrive at the following formula
det(T) di1m = B m ki n2i +2ki ki1+1 = m ki n2i+j2ki ki1+1
| | N,KYi=1 Yj=0(cid:18) j (cid:19)! Yi=1Yj=0 (cid:0)n2i+ki2ki(cid:1)
 
(cid:0) (cid:1)
1
det(T) di1m = m ki k1!(n21 +k1)! ki+1 = m ki j kij+1
| | i=1 j=0 j!(n21 +2k1 −j)!! i=1 j=1(cid:18)n21 +2k1 −j +1(cid:19)
Y Y YY
Applying the trivial bounds ( 1 )j k1!(n21+k1)! and j ki we have
n2i+2ki ≤ j!(n21+2k1−j)! n21+2k1−j+1 ≤ n2i+ki+1
m ki m ki
1 2 1 ki 2
det(T) dim
ni +2k ≤ | | ≤ ni +k +1
i=1(cid:18) 2 i(cid:19) i=1(cid:18) 2 i (cid:19)
Y Y
Let us note that
B−1/2 m e.n2i +2ki k2i
N,K ≤ k
i=1(cid:18) i (cid:19)
Y
ki
m 1 2
−1/2 1 k
B det(T) dim e2 1+
N,K | | ≤ ni +1!
i=1 2ki
Y
(cid:4)
We define a class of Π-harmonic polynomials which turns out to be very useful. Assume
y ,...,y is an orthonormal basis of H . Then let v Sn1−1 Snm−1, q =
1 M N,K v
∈ × ··· ×
M y (v)y . We observe q H moreover f,q = f(v) for all f H . This special
i=1 i i v ∈ N,K h vi ∈ N,K
polynomial q is called the zonal harmonic corresponding to the vector v on H . The
v N,K
P
following lemma states basic properties of zonal harmonics:
10 ALPEREN A. ERGU¨R
Lemma 2.12.
(1) q (w) = q (v) for all v,w S.
v w
∈
(2) q (v) = q (T(v)) for all v,w S and T O(n ) O(n ) O(n ) O(n ).
w T(w) 1 2 3 m
∈ ∈ × × ×···×
(3) q (v) = q 2 = dimH .
v v N,K
k k
(4) q (w) dimH .
v N,K
| | ≤
Proof. (1) Let e ,...,e be an orthonormal basis for H . Then
1 l N,K
q = l e ,q e = m e (v)e . So q (w) = e (v)e (w) = q (v).
v i=1h i vi i i=1 i i v i i w
(2) For p H we have
N,K
P ∈ P P
p(T(w)) = p T,q = p(T(x))q (x)dσ(x) = p(x)q (T−1(x))dσ(x)
w w w
h ◦ i
Z Z
S S
Since the zonal harmonic is unique we deduce that q T−1 = q .
w T(w)
◦
(3) By the notation of (1) and using (2) afterward
q (v) = q ,q = e (v)e , e (v)e = e (v) 2
v v v Π i i i i Π i
h i h i | |
X m X X
q (v) = q (v)dσ(v) = e (v) 2dσ(v) = m = dimH
v v i N,K
| |
Z Z i=1
S S X
(4) q ,q = q (w) q q = dimH . Thus q (w) dimH = q (v).
v w v v w N,K v N,K v
(cid:4)|h i| | | ≤ k kk k | | ≤
Now we define, for any vector v S, the polynomial p = rK−αq where q is
∈ v α∈KK v,α v,α
the zonal harmonic corresponding to v in H . Let f = max f(v) . We observe p
N,α ∞ v∈S v
k k P | |
inherits properties from zonal harmonics:
Lemma 2.13. (1) f(v) = f,p for everyf P .
v N,K
h i ∈
(2) p 2 = p ,p = q ,q = dimH = dimP
v v v v,α v,α N,α N,K
k k h i h i
(3) f(v) = f,p f p and thus kfk∞ p = dimP .
| | |h vi| ≤Pk kk vk P kfk ≤ k vk N,K
(4) p (w) p (v) for all w S and kpvk∞ = dimP .
| v | ≤ v ∈ kpvk N,Kp
The third property turns out to be a characterizpation of the polynomials p :
v
Lemma 2.14. Let f P be such that kfk∞ kgk∞ for all g P . Then f is a constant
∈ N,K kfk ≥ kgk ∈ N,K
multiple of p for some v.
v
Proof. Assume f = f(v) = C and let T = g P : g(v) = C . We observe for g T,
∞ N,K
k k { ∈ } ∈
g C.
∞
k k ≥
Using the assumption on f we deduce g f for all g T. Thus f is the shortest
k k ≥ k k ∈
form on the hyperplane. We also observe g(v) = g,p from Lemma 2.13. This proves f to
v
be a constant multiple of p . (cid:4) h i
v
Let us consider the hyperplane L := q P : q(v) = C = q P : q,p = C .
C N,K N,K v
{ ∈ } { ∈ h i }
We define SO(v) := g SO(n ) SO(n ) : g(v) = v . Now observe that L is fixed
1 m C
under SO(v) action.{Th∈is implies×p··i·s×fixed under SO(v) a}ction and thus, for every c R
v
and M =: x Rn : x,v = c , p is constant on M . This implies p (w) = q ( v,w )∈for
c v c v N,K
{ ∈ h i } h i
some univariate polynomial q . This q is the Gegenbauer or ultraspherical polynomial
N,K N,K