Iterated claws have real-rooted genus polynomials

0.36 MB·English

by  J. L. Gross

ITERATED CLAWS HAVE REAL-ROOTED GENUS POLYNOMIALS
JONATHAN L. GROSS, TOUFIK MANSOUR, THOMAS W. TUCKER,
AND DAVID G.L. WANG
5 Abstract. We prove that the genus polynomials of the graphs called iterated claws
1 are real-rooted. This continues our work directed toward the 25-year-old conjec-
0
ture that the genus distribution of every graph is log-concave. We have previously
2
established log-concavity for sequences of graphs constructed by iterative vertex-

n
amalgamation or iterative edge-amalgamation of graphs that satisfy a commonly
a
observable condition on their partitioned genus distributions, even though it had
J
been proved previously that iterative amalgamation does not always preserve real-
5
2 rootedness of the genus polynomial of the iterated graph. In this paper, the iterated
topological operations are adding a claw and adding a 3-cycle, rather than vertex-

] or edge-amalgamation. Our analysis here illustrates some advantages of employing a
O
matrix representation of the transposition of a set of productions.
C
.
h
t
a
m
Introduction
[ 1.

1
Graphs are implicitly taken to be connected. Our graph embeddings are cellular
v
5 and orientable. For general background in topological graph theory, see [1, 7]. Prior
0
acquaintance with the concepts of partitioned genus distribution (abbreviated here as
1
6 pgd) and production (e.g., [8, 11]) is prerequisite to reading this paper. Subject to this
0 prerequisite, the exposition here is intended to be accessible both to graph theorists
.
1 and to combinatorialists.
0
The genus distribution of a graph G is the sequence g (G), g (G), g (G), ...,
5 0 1 2
1 where g (G) is the number of combinatorially distinct embeddings of G in the ori-
i
:
v entable surface of genus i. A genus distribution contains only finitely many positive
i
X numbers, and there are no zeros between the first and last positive numbers. The
r genus polynomial is the polynomial
a
Γ (z) = g (G)+g (G)z +g (G)z2 +...
G 0 1 2
We say that a sequence A = (a )n is nonnegative if a ≥ 0 for all k. An
k k=0 k
element a is said to be an internal zero of A if there exist indices i and j with
k
i < k < j, such that a a (cid:54)= 0 and a = 0. If a a ≤ a2 for all k, then A is said to
i j k k−1 k+1 k
be log-concave. If there exists an index h with 0 ≤ h ≤ n such that
a ≤ a ≤ ··· ≤ a ≤ a ≥ a ≥ ··· ≥ a ,
0 1 h−1 h h+1 n
2010 Mathematics Subject Classification. 05A15, 05A20, 05C10.
1
2 J.L. GROSS, T. MANSOUR, T.W. TUCKER, AND D.G.L. WANG
then A is said to be unimodal. It is well-known that any nonnegative log-concave
sequence without internal zeros is unimodal, and that any nonnegative unimodal se-
quence has no internal zeros. A prior paper [5] by the present authors provides addi-
tional contextual information regarding log-concavity and genus distributions.
For convenience, we sometimes abbreviate the phrase “log-concave genus distribu-
tion” as LCGD. Proofs that closed-end ladders and doubled paths have LCGDs [2]
were based on closed formulas for their genus distributions. Proof that bouquets have
LCGDs [6] was based on a recursion.
Stahl [12] used the term “H-linear” to describe chains of graphs obtained by amal-
gamating copies of a fixed graph H. He conjectured that a number of “H-linear”
families of graphs have genus polynomials with nonpositive real roots, which implies
the log-concavity of their sequences of coefficients, by Newton’s theorem. Although
it was shown [14] that the genus polynomials of some such families do indeed have
real roots, Stahl’s conjecture of real-rootedness for W -linear graphs (where W is the
4 4
4-wheel) was disproved by Liu and Wang [9].
Ourpreviouspaper[5]proves, nonetheless, thatthegenusdistributionofeverygraph
in the W -linear sequence is log-concave. Thus, even though Stahl’s proposed approach
4
to log-concavity via roots of genus polynomials is sometimes infeasible, [5] does support
Stahl’s expectation that chains of copies of a graph are a relatively accessible aspect
of the general LCGD problem. Moreover, Wagner [14] has proved the real-rootedness
of the genus polynomials for a number of graph families for which Stahl made specific
conjectures of real-rootedness.
Furthermore, we shall see here that Stahl’s method of representing what we have
elsewhere presented as a transposition of a production system for a surgical operation
on graph embeddings as a matrix of polynomials can simplify a proof, that a family of
graphs has log-concave genus distributions.
The Sequence of Iterated Claws
2.
Let the rooted graph (Y ,u ) be isomorphic to the dipole D , and let the root u be
0 0 3 0
either vertex of D . For n = 1,2,..., we define the iterated claw (Y ,u ) to be the
3 n n
graph obtained the following surgical operation:
Newclaw: Subdivide each of the three edges incident on the root ver-
tex u of the iterated claw (Y ,u ), and then join the three new
n−1 n−1 n−1
vertices obtained thereby to a new root vertex u .
n
Figure 2.1 illustrates the graph (Y ,u ).
3 3
ThegraphK iscommonlycalledaclaw graph, whichaccountsforournameiterated
1,3
claw. The notation Y reflects the fact that a claw graph looks like the letter Y. We
n
∼
observe that Y = K . A recursion for the genus distribution of the iterated claw
1 3,3
graphs is derived in [4]. We observe that, whereas all of Stahl’s examples [12] of graphs
with log-concave genus distributions are planar, the sequence of iterated claws has
rising minimum genus.
ITERATED CLAWS HAVE REAL-ROOTED GENUS POLYNOMIALS 3
x
x x 2
1
0
y
y0 u y1 u2 2 u3
u 1
0 z
2
z
1
z
0
Figure 2.1. The rooted graph (Y ,u ).
3 3
Wehaveseeninpreviousstudiesofgenusdistribution(especially[3])thatthenumber
of productions and simultaneous recursions rises rapidly with the number of roots and
the valences of the roots. The surgical operation newclaw is designed to circumvent
this problem.
For a single-rooted iterated claw (Y ,u ), we can define three partial genus dis-
n n
tributions, also called partials. Let
a = the number of embeddings Y → S such that
n,i n i
three different fb-walks are incident on the root u ;
n
b = the number of embeddings Y → S such that exactly
n,i n i
two different fb-walks are incident on the root u ;
n
c = the number of embeddings Y → S such that
n,i n i
one fb-walk is incident three times on the root u .
n
We also define the generating functions
∞
(cid:88)
A (z) = a zi
n n,i
i=0
∞
(cid:88)
B (z) = b zi
n n,i
i=0
∞
(cid:88)
C (z) = c zi
n n,i
i=0
A listing of the non-zero values of all the partials for every genus i is called a par-
titioned genus distribution. Clearly, the full genus distribution is the sum of the
partials. That is, for i = 0,1,2,..., we have
g (Y ) = a + b + c
i n n,i n,i n,i
and
Γ (z) = A (z) + B (z) + C (z)
Yn n n n
We define g = g (Y ).
n,i i n
4 J.L. GROSS, T. MANSOUR, T.W. TUCKER, AND D.G.L. WANG
Theorem 2.1. For n > 1, the effect on the pgd of applying the operation newclaw to
the iterated claw (Y ,u ) corresponds to the following system of three productions:
n−1 n−1
(2.1) a −→ 12b +4c
i i+1 i+2
(2.2) b −→ 2a +12b +2c
i i i+1 i+1
(2.3) c −→ 8a +8c
i i i+1
Proof. This is Theorem 4.5 of [4]. (cid:3)
Corollary 2.2. For n > 1, the effect on the pgd of applying the operation newclaw to
the iterated claw (Y ,u ) corresponds to the following recurrence relations:
n−1 n−1
(2.4) a = 2b + 8c
n,i n−1,i n−1,i
(2.5) b = 12a + 12b
n,i n−1,i−1 n−1,i−1
(2.6) c = 4a + 2b + 8c
n,i n−1,i−2 n−1,i−1 n−1,i−1
Proof. The recurrence system (2.4), (2.5), (2.6) is reasonably described as a transpo-
sition of the production system (2.1), (2.2), (2.3). (cid:3)
It is convenient to express such a recurrence system in matrix form:
(2.7) V(Y ) = M(z)·V(Y )
n n−1
with the initial column vector
   
A(Y ) 2
0
(2.8) V(Y0) = B(Y0) = 0
C(Y ) 2z
0
and the production matrix
 
0 2 8
(2.9) M(z) = 12z 12z 0
4z2 2z 8z
Proposition 2.3. The column vector V(Y ) is the product of the matrix power Mn(z)
n
with the column vector V(Y ).
0
Corollary 2.4. The column vector V(Y ) is the product of the matrix power Mn+1(z)
n
with the (artificially labeled) column vector
 
0
V(Y−1) =  0 
1/4
Corollary 2.5. To prove that every iterated claw has an LCGD, it is sufficient to prove
that the sum of the third column of the matrix Mn(z) is a log-concave polynomial.
ITERATED CLAWS HAVE REAL-ROOTED GENUS POLYNOMIALS 5
Remark. Partitioned genus distributions and recursion systems for pgds were first
usedbyFurst,Gross,andStatman[2]. Stahl[12]wasfirsttoemployamatrixequivalent
of a production system to investigate log-concavity. A presentation by Mohar [10] has
called more recent attention to the matrix equivalent and its properties.
Characterizing Genus Polynomials for Iterated Claws
3.
In this section, we investigate some properties of the genus polynomials of iterated
claws. Corollary 2.5 leads us to focus on the sum of the third column of the matrix
Mn(z), which is expressible as (1,1,1)Mn(z)(4V(Y )), which implies that it equals
−1
4 times the genus polynomial of the iterated claw Y . Theorem 3.1 formulates a gen-
n−1
erating function f(z,t) for this sequence of sums, and Theorem 3.2 uses the generating
function to construct an expression for the genus polynomials from which we establish
interlacing of roots in Section 4.
Theorem 3.1. The generating function f(z,t) = (cid:80) (1,1,1)Mn(z)(4V(Y ))tn for
n≥0 −1
the sequence of sums of the third column of Mn(z) has the closed form
1+(8−12z)t−24zt2
(3.1) f(z,t) = .
1−20zt+8z(8z −3)t2 +384z3t3
Proof. Let (p ,q ,rq ) = (1,1,1)Mn for all n ≥ 0. Then
n n n
(3.2) (p ,q ,r ) = (p ,q ,r )M(z)
n+1 n+1 n+1 n n n
= (12zq +4z2r , 2p +12zq +2zr , 8p +8zr ).
n n n n n n n
The third coordinate of Equation (3.2) implies that
1
(3.3) p = (r −8zr ).
n n+1 n
8
By combining (3.3) with the first coordinate of (3.2) we obtain
1
(3.4) q = (r −8zr −32z2r ).
n n+2 n+1 n
96z
The second coordinate of (3.2) yields
(3.5) q = 2p +12zq +2zr
n+1 n n n
Substituting (3.3) and (3.4) into (3.5) leads to the recurrence relation
(3.6) r = 20zr +8z(3−8z)r −384z3r
n n−1 n−2 n−3
with
r = 1,
0
(3.7) r1 = 8+8z,
r = 160z +96z2.
2
By multiplying Recurrence (3.6) by tn and summing over all n ≥ 0, we obtain Gener-
ating Function (3.1). (cid:3)
6 J.L. GROSS, T. MANSOUR, T.W. TUCKER, AND D.G.L. WANG
Theorem 3.2. The genus polynomial of the iterated claw Y is given by
n
(1,1,1)Mn+1(z)V(Y ) = 2n−1(h (z)+2(2−3z)h (z)−6zh (z)),
−1 n+1 n n−1
where
(cid:88) (cid:18)j +i (cid:19)(cid:18)j +i (cid:19)(cid:18)j +i (cid:19) √ √
h (z) = 1 2 3 (1+ 3)i2(1− 3)i33j+i1(2z)n−j.
n
i i i
1 2 3
2j+i1+i2+i3=n
Proof. By Theorem 3.1, we have
(cid:88) 1+(8−12z)t−24zt2
f(z,t) = (1,1,1)Mn(4V(Y ))tn = .
0 1−20zt+8z(8z −3)t2 +384z3t3
n≥0
Thus,
1+(4−3z)t−3zt2
f(z/2,t/2) =
1−5zt+z(4z −3)t2 +6z3t3
1+(4−3z)t−3zt2
=
(1−2zt−2z2t2)(1−3zt)−3zt2
(cid:88) (1+(4−3z)t−3zt2)3jzjt2j
= √ √ .
(1−3zt)j+1(1+ 3zt)j+1(1− 3zt)j+1
j≥0
Using the combinatorial identity (1−at)−m = (cid:80) (cid:0)m−1+j(cid:1)ajtj, and then finding the
j≥0 j
coefficient of tn, we derive the equation
(1,1,1)Mn(z/2)V(Y ) = 2n−2(h (z)+2(2−3z)h (z)−6zh (z)),
0 n n−1 n−2
which, by Corollary 2.4, completes the proof. (cid:3)
Theorem 3.2 provides an explicit expression for the genus polynomial Γ (z). It is
Yn
easy to see that Γ (z) = r /4, where r is defined in the proof of Theorem 3.1. In
Yn n+1 n
terms of Γ (z), the recurrence relation (3.6) becomes
Yn
(3.8) Γ (z) = 20zΓ (z)+8z(3−8z)Γ (z)−384z3Γ (z).
Yn Yn−1 Yn−2 Yn−3
Let g be the coefficient of zi in Γ (z). The following table of values of g for
n,i Yn n,i
n ≤ 4 is derived in [4].
g i = 0 1 2 3 4 5
n,i
n = 0 2 2 0 0 0 0
1 0 40 24 0 0 0
2 0 48 720 256 0 0
3 0 0 1920 11648 2816 0
4 0 0 1152 52608 177664 30720
Denote by P the set of polynomials of the form (cid:80)t a zk, where a is a positive
s,t k=s k k
integer for any s ≤ k ≤ t. The above table suggests that Γ (z) ∈ P for
Yn (cid:98)(n+1)/2(cid:99),n+1
n ≤ 4. Now we show it holds true in general.
ITERATED CLAWS HAVE REAL-ROOTED GENUS POLYNOMIALS 7
Theorem 3.3. For all n ≥ 0, the polynomial Γ (z) ∈ P . Moreover, we
Yn (cid:98)(n+1)/2(cid:99),n+1
have the leading coefficient
(cid:98)(n+1)/2(cid:99)(cid:18) (cid:19)
(cid:88) n+2
(3.9) g = 4n 3k,
n,n+1
2k +1
k=0
and, for any number i such that (cid:98)(n+1)/2(cid:99)+1 ≤ i ≤ n, we have
(3.10) g > 11g .
n,i n−1,i−1
Proof. We see in the table above that Γ (z) ∈ P and that Equation (3.9)
Yn (cid:98)(n+1)/2(cid:99),n+1
and Inequality (3.10) are true, for n ≤ 4. Now suppose that n ≥ 5. For convenience, let
g = 0 for all i < 0. We can also take g = 0 for i > k +1, by induction using (3.8),
k,i k,i
for k < n. From Recurrence (3.8) and the induction hypothesis, we have
(3.11) g = 20g +24g −64g −384g , n ≥ 3.
n,i n−1,i−1 n−2,i−1 n−2,i−2 n−3,i−3
For i > n+1, the induction hypothesis implies that each of the four terms on the
right side of Recurrence (3.11) is zero-valued. So the degree of Γ (z) is at most n+1.
Yn
Let s = g . Taking i = n+1 in (3.11), we get
i i,i+1
(3.12) s = 20s −64s −384s ,
n n−1 n−2 n−3
with the initial values s = 2, s = 24, s = 256. The above recurrence can be solved by
0 1 2
a standard generating function method, see [15, p.8]. In practice, we use the command
rsolve in the software Maple and get the explicit formula directly as
(cid:18) (cid:19)
(cid:88) n+2
s = 4n 3k.
n
2k +1
k≥0
It follows that g > 0. Hence the degree of Γ (z) is exactly n+1.
n,n+1 Yn
Similarly, for i < (cid:98)(n + 1)/2(cid:99), the four terms on the right side of (3.11) are zero-
valued, so the minimum genus of Y is at least (cid:98)(n+1)/2(cid:99). Moreover, applying (3.11)
n
with i = (cid:98)(n+1)/2(cid:99) and using the induction hypothesis g = 0 for all i < (cid:98)(k+1)/2(cid:99)
k,i
with k < n, we find the first term is positive for n odd and zero for n even, the second
term is always positive, and the third and fourth terms are always zero. In other words,
g = 20g +24g ≥ 24g > 0.
n,(cid:98)(n+1)/2(cid:99) n−1,(cid:98)(n+1)/2(cid:99)−1 n−2,(cid:98)(n+1)/2(cid:99)−1 n−2,(cid:98)(n+1)/2(cid:99)−1
This confirms the minimum genus of Y is exactly (cid:98)(n+1)/2(cid:99).
n
Now consider i such that (cid:98)(n + 1)/2(cid:99) + 1 ≤ i ≤ n. By (3.11), and using (3.10)
inductively, we deduce
g = 11g +24g +(9g −64g −384g )
n,i n−1,i−1 n−2,i−1 n−1,i−1 n−2,i−2 n−3,i−3
> 11g +24g +(35g −384g )
n−1,i−1 n−2,i−1 n−2,i−2 n−3,i−3
> 11g +24g +g
n−1,i−1 n−2,i−1 n−3,i−3
≥ 11g .
n−1,i−1
So Inequality (3.10) holds true. It follows that g > 0. Hence
n,i
Γ (z) ∈ P .
Yn (cid:98)(n+1)/2(cid:99),n+1
This completes the proof. (cid:3)
8 J.L. GROSS, T. MANSOUR, T.W. TUCKER, AND D.G.L. WANG
Corollary 3.4. The sequence of coefficients of Γ (z) has no internal zeros.
Yn
Proof. This special case of the familiar and easily proved “interpolation theorem” of
topological graph theory also follows directly from Theorem 3.3. (cid:3)
Genus Polynomials for Iterated Claws are Real-Rooted
4.
Our goal in this section is to establish in Theorem 4.2 the real-rootedness of the
genus polynomials Γ (z) of the iterated claws, via an associated sequence W (z) of
Yn n
normalized polynomials. It follows from this real-rootedness, by Newton’s theorem
(e.g., see [13], Theorem 2), that the genus polynomials for iterated claws are log-
concave.
To proceed, we “normalize” the polynomials Γ (z) by defining
Yn
(4.1) W (z) = z−(cid:98)(n+1)/2(cid:99)Γ (z),
n Yn
so that W (z) starts from a non-zero constant term, and has the same non-zero roots
n
as Γ (z). We use the symbol d to denote the degree of W (z), that is,
Yn n n
(cid:22) (cid:23) (cid:24) (cid:25)
n+1 n+1
(4.2) d = degW (z) = (n+1)− = .
n n
2 2
By Theorem 3.3, we have W (z) ∈ P . Substituting (4.1) into the recurrence rela-
n 0,dn
tion (3.8), we derive
(cid:40)
20zW (z)+8(3−8z)W (z)−384z2W (z), if n is even,
n−1 n−2 n−3
(4.3) W (z) =
n
20W (z)+8(3−8z)W (z)−384zW (z), if n is odd,
n−1 n−2 n−3
with the initial polynomials
W (z) = 2(1+z),
0
(4.4) W (z) = 8(5+3z),
1
W (z) = 16(3+45z +16z2).
2
Let P denote the union ∪ P = ∪ {(cid:80)n a zk|a ∈ Z+}. Lemma 4.1 is an
n≥0 0,n n≥0 k=0 k k
elementary consequence of the intermediate value theorem.
Lemma 4.1. Let P(x),Q(x) ∈ P. Suppose that P(x) has roots x < x < ··· < x ,
1 2 degP
and that Q(x) has roots y < y < ··· < y . If degQ − degP ∈ {0,1} and if the
1 2 degQ
roots interlace so that
x < y < x < y < ··· ,
1 1 2 2
then
(4.5) (−1)i+degPP(y ) > 0 for all 1 ≤ i ≤ degQ,
i
(4.6) (−1)j+degQQ(x ) < 0 for all 1 ≤ j ≤ degP.
j
ITERATED CLAWS HAVE REAL-ROOTED GENUS POLYNOMIALS 9
Proof. Since P(x) is a polynomial with positive coefficients, we have
(4.7) (−1)degPP(−∞) > 0.
We suppose first that degP(x) is odd, and we consider the curve P(x). We see that
Inequality (4.7) reduces to P(−∞) < 0. Thus, the curve P(x) starts in the lower half
plane and intersects the x-axis at its first root, x . From there, the curve P(x) proceeds
1
without going below the x-axis, until it meets the second root, x . Since x < y < x ,
2 1 1 2
we recognize that (4.5) holds for i = 1, i.e.,
(4.8) P(y ) > 0.
1
Afterpassingthroughx , thecurveP(x)staysbelowthex-axisuptothethirdroot,x .
2 3
It is clear that the curve P(x) continues going forward, intersecting the x-axis in this
alternating way. It follows from this alternation that
(4.9) P(y )P(y ) < 0 for all 1 ≤ k ≤ degQ−1.
k k+1
From (4.8) and (4.9), we conclude that (4.5) holds for all 1 ≤ i ≤ degQ, when degP(x)
is odd.
We next suppose that degP(x) is even. In this case, we can draw the curve P(x)
so that it starts in the upper half plane, first intersects the x-axis at x , then goes
1
below the axis up to x , and continues alternatingly. Therefore the sign-alternating
2
relation (4.9) still holds. Since P(y ) < 0 when degP(x) is even, we have proved (4.5).
1
It is obvious that Inequality (4.6) can be shown along the same line. This completes
the proof of Lemma 4.1. (cid:3)
Now we proceed with our main theorem on the genus polynomial of iterated claws.
Theorem 4.2. For every n ≥ 0, the polynomial W (z) is real-rooted. Moreover, if the
n
roots of W (z) are denoted by x < x < ···, then we have the following interlacing
k k,1 k,2
properties:
(i) for every n ≥ 2, the polynomial W (z) has one more root than W (z), and
n n−2
the roots interlace so that
x < x < x < x < ··· < x < x < x ;
n,1 n−2,1 n,2 n−2,2 n,dn−1 n−2,dn−1 n,dn
(ii) for every n ≥ 1, the polynomial W (z) has either one more (when n is even) or
n
the same number (when n is odd) of roots as W (z), and the roots interlace
n−1
so that
x < x < x < x < ··· < x < x when n even;
n,1 n−1,1 n,2 n−1,2 n−1,dn−1 n,dn
and
x < x < x < x < ··· < x < x when n odd.
n,1 n−1,1 n,2 n−1,2 n,dn n−1,dn
Proof. From the initial polynomials (4.4), it is easy to verify Theorem 4.2 for n ≤ 2.
We suppose that n ≥ 3 and proceed inductively.
For every k ≤ n − 1, we denote the roots of W (z) by x < x < ··· < x .
k k,1 k,2 k,d
k
For convenience, we define x = −∞ and x = 0, for all k ≤ n − 1. To clarify
k,0 k,d +1
k
the interlacing properties, we now consider the signs of the function W (z) at −∞
m
10 J.L. GROSS, T. MANSOUR, T.W. TUCKER, AND D.G.L. WANG
and at the origin, for any m ≥ 0. Since W (z) is a polynomial of degree d , with all
m m
coefficients non-negative, we deduce that
(4.10) (−1)dmW (−∞) > 0.
m
Having the constant term positive implies that
(4.11) W (0) = g > 0.
m n,0
By the intermediate value theorem and Inequality (4.10), for the polynomial W (z)
n
to have d = degW (z) distinct negative roots and for Part (i) of Theorem 4.2 to hold,
n n
it is necessary and sufficient that
(4.12) (−1)dn+jW (x ) > 0 for 1 ≤ j ≤ d +1.
n n−2,j n−2
In fact, for j = d +1, Inequality (4.12) becomes
n−2
(4.13) (−1)dn+dn−2+1W (0) > 0.
n
By (4.11), Inequality (4.13) holds if and only if d +d is odd, which is true since
n n−2
(cid:24) (cid:25) (cid:24) (cid:25) (cid:24) (cid:25)
n+1 n−1 n−1
d +d = + = 2 +1.
n n−2
2 2 2
Now consider any j such that 1 ≤ j ≤ d . We are going to prove (4.12). We will
n−2
use the particular indicator function I , which is defined by
even
(cid:40)
1, if n is even,
I (n) =
even
0, if n is odd.
Note that x is a root of W (z). By Recurrence (4.3), we have
n−2,j n−2
(cid:16) (cid:17)
(4.14) W (z ) = xIeven(n) 20W (x )−384x W (x ) .
n n−2,j n−2,j n−1 n−2,j n−2,j n−3 n−2,j
Since x < 0, the factor xIeven(n) contributes (−1)n+1 to the sign of the right hand
n−2,j n−2,j
side of (4.14). On the other hand, it is clear that the sign of the parenthesized factor
can be determined if both the summands 20W (x ) and −384x W (x )
n−1 n−2,j n−2,j n−3 n−2,j
have the same sign. Therefore, Inequality (4.12) holds if
(4.15) (−1)dn+j+n+1W (x ) > 0,
n−1 n−2,j
(4.16) (−1)dn+j+n+1W (x ) > 0.
n−3 n−2,j
Bytheinductionhypothesisonpart(ii)ofthistheorem, wecansubstituteP = W
n−1
and Q = W into Lemma 4.1. Then Inequality (4.5) gives
n−2
(4.17) (−1)dn−1+jW (x ) > 0.
n−1 n−2,j
Thus, Inequality (4.15) holds if and only if the total power
(cid:24) (cid:25) (cid:24) (cid:25)
n+1 n
d +j +n+1+d +j = + +n+2j +1
n n−1
2 2