Extension of the two-variable Pierce-Birkhoff conjecture to generalized polynomials

0.25 MB·English

by  Charles N. Delzell

EXTENSION OF THE TWO-VARIABLE PIERCE-BIRKHOFF
CONJECTURE TO GENERALIZED POLYNOMIALS
0
1 CHARLESN.DELZELL
0
2 In honor of MelvinHenriksen’s80th birthday

n
a Abstract. [English version:] Let h : Rn → R be a continuous, piecewise-
J
polynomialfunction. ThePierce-Birkhoffconjecture(1956)isthatanysuchh

0 is representable in the form supiinfjfij, for some finite collection of polyno-
3 mialsfij ∈R[x1,...,xn]. (Asimpleexampleish(x1)=|x1|=sup{x1,−x1}.)
In1984,L.Mah´eand,independently, G.Efroymson,provedthisforn≤2;it

] remainsopenforn≥3. Inthispaperweproveananalogousresultfor“gener-
G
alizedpolynomials”(alsoknown assignomials), i.e.,wherethe exponents are
A allowed to be arbitrary real numbers, and not just natural numbers; in this
. version,werestricttothepositiveorthant,whereeachxi>0. Asbefore,our
h methods workonlyforn≤2.
at [French version:] En 1984, L. Mah´e, et ind´ependammant G. Efroymson,
m ont prouv´e le cas ou` n ≤ 2 de la conjecture de Pierce-Birkhoff (1956) : une
fonctionh:Rn→Rcontinuepolynomialeparmorceauxpeuts’´ecrirecomme
[ supiinfjfij, pour une collection finie de polynoˆmes fij ∈R[x1,...,xn]. (Un
1 exemplesimpleesth(x1)=|x1|=sup{x1,−x1}.)Laconjectureresteouverte
pour n≥3. Dans cet article, nous prouvons (encore pour n≤2) un r´esultat
v
analogue pour ≪polynoˆmes g´en´eralis´es≫, ou` les exposants peuvent ˆetre des
8
nombres r´eels arbitraires, et non pas seulement des nombres naturels; dans
3
0 cette version,nouslimitonsledomaine`al’orthantpositif,ou` chaquexi>0.
0
.
2
0
1. Generalized polynomial functions
0
1 and generalized semialgebraic sets
:
v We write R =[0,∞) and R =(0,∞), endowed with the usual, order topol-
+ ++
i ogy. And the Cartesian product, R2 := R ×R will be endowed with the
X ++ ++ ++
usual, Euclidean topology.
r
a
Definition 1.1. A generalized polynomial function a(x,y) of two variables is a
function a:R2 →R of the form
++
a:=a(x,y):=c xα1,1yα1,2 +c xα2,1yα2,2 +···+c xαm,1yαm,2, (1.1.1)
1 2 m
where m ∈ N := {0,1,2,...}, the “coefficients” c of a are nonzero elements of R,
i
andthe (binary)“exponents”α :=(α ,α )ofaaredistinctelements ofR2. We
i i,1 i,2
Date:November21,2009.
2000 Mathematics Subject Classification. Primary 14P15; secondary 03C64, 06B25, 26B99,
26C05.
Key words and phrases. real analytic geometry, Pierce-Birkhoff,signomial, piecewise-polyno-
mial,continuous, f-ring,o-minimal.
Toappear inAnnales de la Facult´e des Sciences de Toulouse. Theresultsinthispaper were
first presented at the Conference on Ordered Rings (“Ord007”), at Louisiana State University,
BatonRouge,Louisiana,USA,April25–28, 2007: http://www.math.lsu.edu/∼madden/Ord007.
1
2 CHARLESN.DELZELL
write R[R2] for the ring (actually, it is a group ring) of all generalized polynomial
functions a:R2 →R.
++
Thus,generalizedpolynomialfunctions(sometimescalled“signomial”functions)
oftwovariablescanbedefined,roughly,as“realpolynomialfunctionsonR2 with
++
arbitrary real exponents.” A simple example is a(x,y)=y−xπ.
Generalized polynomial functions of two variables are clearly real analytic on
R2 .
++
See [Delzell, 2008] for background on the general properties and the history
of generalized polynomials (in any number of variables), and some motivation for
studying them.
Definition 1.2. We call a subset A ⊆ R2 a generalized semialgebraic set, or a
++
semisignomial set,ifit isofthe form J S ,where J ∈N andeachS is a “basic
j=1 j j
semisignomial” set, i.e., one of the form
S
S ={(x,y)∈R2 |f (x,y)=0, g (x,y)>0,...,g (x,y)>0}, (1.2.1)
j ++ j j,1 j,Kj
where each K ∈N and the f and g are generalized polynomials.
j j jk
(Recallthat ordinarysemialgebraicsubsets ofR2 orRn aredefined analogously,
but with the f and g being (ordinary) polynomials.)
j jk
2. Piecewise generalized polynomial functions
Definition 2.1. We call a function h(x,y) : R2 → R a piecewise generalized
++
polynomial function oftwovariablesifthereexistg ,...,g ∈R[R2](1.1)suchthat
1 l
the subsets
A :={(x,y)∈R2 |h(x,y)=g (x,y)} (2.1.1)
i ++ i
are generalized semialgebraic and cover R2 , i.e., R2 = A .
++ ++ i i
We may, and shall, assume that the g are distinct.
i
S
Example 2.2.
y
6
(1,1)s
y−xπ if y ≥xπ, y =xπ
h(x,y):= h=y−xπ
(0 if y <xπ.
h=0
-
0 x
The following, technical lemma will not be needed until Proposition 4.8 and
Lemma 5.3 below, and can be skipped on a first reading. In it, for any set A in
R2 , we shall write A◦ for the interior of A.
++
Lemma 2.3. Let A ,...,A be as in (2.1).
1 l
l
(1) A◦ is dense in R2 .
i ++
i=1
[
(2) A◦∩A◦ =∅ for i6=j.
i j
(3) If h is continuous, then each A is closed, whence A◦ ⊆A .
i i i
TWO-VARIABLE PIERCE-BIRKHOFF FOR GENERALIZED POLYNOMIALS 3
l
(4) If h is continuous, then A◦ =R2 \ A◦∩A◦ .
i ++ i j
i=1 1≤i<j≤l
(5) Suppose h is continuous[, and E is a co[nnec(cid:0)ted subse(cid:1)t of R2 such that
++
for each (x,y) ∈ E, the l values g (x,y),g (x,y),...,g (x,y) are distinct. Then
1 2 l
there exists an i ∈ {1,2,...,l} such that E ⊆ A◦ (in particular, such that h = g
i i
throughout E). This i is unique in case E 6=∅.
Proof. (1) By (1.2), A is a combined, but still finite, union of suitable basic
i i
semisignomial sets S as in (1.2.1). Let T be the union of those S for which
j j
f 6≡ 0; thus, T ⊆ Z(FS) := {(x,y) ∈ R2 | F(x,y) = 0}, where F is the product
j ++
ofthosef ’s. R2 \Z(F)isdenseinR2 ,bytheidentitytheoremforrealanalytic
j ++ ++
functions. A fortiori, R2 \T is also dense in R2 . The unionU of the other S ’s
++ ++ j
(viz., those for which f ≡ 0) must contain R2 \T (since T ∪U = A = R2
j ++ i i ++
(2.1)), and so U is also dense in R2 . But A◦ ⊇U.1
++ i i S
(2) If A◦ ∩A◦ 6= ∅, then g would agree with g on a nonempty open set (by
i j i S j
(2.1.1)),andhenceonallofR2 (againbytheidentitytheorem),contradictingthe
++
distinctness of the g in (2.1).2
i
(3) Obvious.
(4) ⊆. Let (x,y)∈A◦ and suppose j 6=i. It is enough to show that (x,y)∈/ A◦.
i j
There exists an open disk in A about (x,y). In fact, this disk is in A◦, and hence
i i
is disjoint from A◦, by (2) above. Therefore (x,y)∈/ A◦.3
j j
⊇. Suppose (x,y) ∈ R2 \ A◦. For r ∈ R with r ≤ min{x,y}, let B
++ i i ++ r
denote the open disk in R2 of radius r > 0 about (x,y), and let I(r) = {i ∈
++ S
{1,2,...,l} | B ∩A◦ 6= ∅}. Then for every r, |I(r)| ≥ 1, by (1) above. In fact,
r i
I(r) > 1. Otherwise, for some i, A◦ ∩B would be dense in B (by (1) again),
i r r
whence B = A◦ ∩B ⊆ A ∩B (by (3)),4 i.e., B ⊆ A , whence (x,y) ∈ A◦,
r i r i r r i i
contradiction. Now, for any s ∈ R with s < r, I(s) ⊆ I(r); i.e., the finite set
++
I(r) decreases monotonically with r, and yet always has cardinality ≥ 2. Thus,
there exist at least two indices i < j such that for every r ∈ (0,min{x,y}), B
r
meets A◦ and A◦. Therefore (x,y)∈A◦∩A◦.
i j i j
(5) The distinctness hypothesis of (5) can be rephrased as
E∩ (A ∩A )=∅.
i j
i<j
[
A fortiori, E∩ A◦ ∩A◦) = ∅, using (3). By (4), E ⊆ A◦. The existence
i<j i j i i
of the desired i now follows from (2) and the hypotheses that E is connected. The
uniqueness of i iSn cas(cid:0)e E 6=∅ also follows from (2). S (cid:3)
Remark 2.4. In Remark 5.4 below, we shall use (2.3) above to see that when a
piecewise generalized polynomial function h is continuous, each A in (2.1) can
i
automatically be taken to be a generalized semialgebraic set; it is not necessary to
include that condition as a hypothesis in (2.1).
1Infact,S A◦=U. Butwedon’tneedthis.
i i
2Andifgiagreeswithgj onallofR2++,thenthecoefficientsofgiandgj (i.e.,thec’sin(1.1.1)
above)wouldagree,too,by[Delzell,2008, Remark4.3].
3Thishalfoftheproofof(4)doesnotrequirethehypothesis thathbecontinuous.
4Infact,thisinclusionisactuallyanequality.
4 CHARLESN.DELZELL
The setofpiecewise generalizedpolynomialfunctions is closedunder differences
and products, and so forms a ring; it is also closed under pointwise suprema and
infima, and so forms an l-ring under those lattice operations. (This ring is, of
course, even an f-ring.) The continuous functions in this f-ring comprise a sub-f-
ring. (See, e.g., [Birkhoff, et al., 1956] or [Henriksen, et al., 1962] for background
on l-rings and f-rings.)
3. Statement and discussion of the main result
Theorem 3.1 (Main Theorem: The Pierce-Birkhoff conjecture for generalized
polynomials in two variables). If h :R2 →R is continuous and piecewise gener-
++
alized polynomial, then h is a (pointwise) sup of infs of finitely many generalized
polynomial functions; i.e.,
h(x,y)=supinff (x,y) on R2 , (3.1.1)
jk ++
j k
for some finite number of generalized polynomials f . (The converse is easy.)
jk
Example 3.2. For the h in Example 2.2 above, h(x,y)=sup{0, y−xπ}.
The representation of h in the form (3.1.1) makes both the continuity and the
piecewise generalized polynomial character of h obvious.
ForordinarypolynomialsinR[X,Y]andordinarypiecewisepolynomialfunctions
onR2,theanalogofTheorem3.1abovewasfirstprovedbyL.Mah´e[[cite]cite.Mahe
1984Mah´e,1984]andEfroymson(unpublished), independently. Thestatementand
proofs of the Mah´e-Efroymson theorem generalize easily to the situation where R
isreplacedbyanarbitraryrealclosedfieldR (furnished withthe topologyinduced
by the unique ordering on R). But the fact that then the coefficients of the f in
jk
the Mah´e-Efroymson theorem may be taken to lie in the subfield of R generated
by the coefficients of the g defining h (in the analog of (2.1)), was not trivial, and
i
was proved in [Delzell, 1989].
The extension of the Mah´e-Efroymson theorem to functions of three or more
variables (like the extension of (3.1) above) remains unproved and unrefuted; it
is known as the Pierce-Birkhoff Conjecture (first formulated in [Birkhoff, et al.,
1956]).
In our proof of Theorem 3.1 below, we shall make no attempt to indicate which
stepsgeneralizeeasilytothecasewheren>2(thoughmanyofthosestepsdo). The
first reason for this is that the notation is often simpler when n = 2. The second
reason is that, considering the many mathematicians who have tried to prove the
Pierce-Birkhoff Conjecture for n > 2, we now lean toward the opinion that it and
Theorem 3.1 are false for n>2.
In 1987we provedthat for all n≥1 and everyrealclosedfield R, if h:Rn →R
is “piecewise-rational” (i.e., if there are rational functions g ,...,g ∈ R(X) such
1 l
that the sets A :={x∈Rn |g (x) is defined and h(x)=g (x)} are s.a. and cover
i i i
Rn), then there are finitely many f ∈ R(X) and there is a k ∈ R[X ,...,X ]\
jk 1 n
{0} such that for all x ∈ Rn where k(x) 6= 0 (i.e., for “almost all” x ∈ Rn),
each f (x) is defined and h(x) = sup inf f (x); this is true even if h is not
jk j k jk
continuous. This result was announced in [Delzell, 1989, p. 659], and proved in
[[cite]cite.Delzell 1990Delzell, 1990]. Madden gave an “abstract” version of this
result that applies to arbitrary fields (and not just R(X)); see [Madden, 1989].
In [[cite]cite.Delzell 2005Delzell, 2005] we proved an analog of our 1987 result,
TWO-VARIABLE PIERCE-BIRKHOFF FOR GENERALIZED POLYNOMIALS 5
for “generalized piecewise-rational functions” (i.e., functions that are, piecewise,
quotients of generalized polynomial functions).
The rest of this paper will be devoted to the proof of Theorem 3.1. In §4 we
shall develop the necessary one-variable machinery; in §5 we shall deal with the
additional difficulties arising in the two-variable situation.
4. One-variable methods
We imitate Mah´e’s proof as much as possible.
We are given a continuous function
g (x,y) if (x,y)∈A
1 1
. .
h(x,y)= .. .. (4.0.1)
g (x,y) if (x,y)∈A ,
l l
where,asin(2.1),thegiaregeneralizedpolynomialsandtheAi coverR2++. (Recall
from Remark 2.4 above that the A are also, automatically, generalized semialge-
i
braic; but we don’t use this.) As before, we assume the g are distinct.
i
Write each a(x,y)∈R[R2]\{0} (1.1) in the form
a (x)yβ1 +a (x)yβ2 +···+a (x)yβK, (4.0.2)
1 2 K
whereK ≥1, β <···<β ∈R, andeacha is a nonzerogeneralizedpolynomial
1 K i
in x. This representation is unique.
Let A = {g −g | 1 ≤ i < j ≤ l}. Let B be the smallest subset of R[R2]
i j
containing A and closed under the following two operations, for each a(x,y) ∈ B
for which K >1 in (4.0.2):
∂a
a′ := if β =0, and
1
a7→ ∂y (4.0.3)
y−β1a(x,y) if β 6=0 5; and
 1
y
r :=r (x,y)=a(x,y)− ·a′(x,y) if β =0,6 and
a 1
a7→ βK (4.0.4)

a if β 6=0.
 1

Remark 4.1. Suppose no g involves the variable x; i.e., each g is a function of y
i i
alone, and is constant in x. Then the same is, of course, true for each a ∈ A; in
fact, the same is true even for each a∈B, in view of (4.0.3) and (4.0.4).
Lemma 4.2. For each a ∈ B for which K > 1 and β = 0, a′(x,y) and r each
1 a
have exactly K−1 y-terms. Consequently, B is finite.
Proof. Thisisclearfora′(x,y). Forra,observe(a)thattheKthy-termaK(x,y)yβK
in a (4.0.2) is cancelled out by the y-term
y
β a (x,y)yβK−1
K K
β
K
in (cid:0) (cid:1)
y
·a′(x,y), (4.2.1)
β
K
5Thistrick(ofdividingbyyβ1)wasfirstusedbySturm[[cite]cite.Sturm1829Sturm,1829].
6HereweuseβK 6=0,whichfollowsfromβ1=0andK>1.
6 CHARLESN.DELZELL
and (b) that the other y-terms of (4.2.1) involvethe y-exponents β ,...,β , but
1 k−1
with coefficients different from those of the corresponding y-terms of a (since for
each i<K, β /β 6=1). (cid:3)
i K
Lemma 4.3. There exist L ∈ N and γ < γ < ··· < γ ∈ R such that,
1 2 L ++
writing γ = 0 and γ = ∞, for each a ∈ B and for each p ∈ {0,1,...,L}, the
0 L+1
zeros of a(x,y) in the pth vertical half strip H :=(γ ,γ )×R are the graphs
p p p+1 ++
of continuous, monotonic7 “generalized semialgebraic”8 functions y = ξ (x),
a,p,j
j =1,2,...,s (where s:=s(a,p) satisfies 0≤s≤K9) with
(0<)ξ <···<ξ on (γ ,γ ).
a,p,1 a,p,s p p+1
Moreover, ∀a ,a ∈ B, ∀p ≤ L, ∀j ≤ s(a ,p), ∀j ≤ s(a ,p), throughout
1 2 1 1 2 2
(γ ,γ )⊆R , only one of the following three relations holds:
p p+1 ++
ξ <ξ ,
a1,p,j1 a2,p,j2
ξa1,p,j1 =ξa2,p,j2, or (4.3.1)
ξ >ξ .
a1,p,j1 a2,p,j2
Lemma4.3anditsCorollary4.5areillustratedinFigure1,whichalsoshowsthe
stackof open connected sets D ,D ,D whose union is a dense open subset of
2,1 2,2 2,3
H (looking ahead to (4.5) below).
2
Proof. Miller [[cite]cite.Miller 1994Miller, 1994] considered a class of functions f :
Rn → R that properly contains the class of (extensions by 0 to Rn of) gener-
alized polynomial functions. Specifically, he considered terms built up (in a for-
mal language) from variable symbols x ,x ,... and from constants in R by the
1 2
usualoperationsymbols +, −, and · , together with the class of operationsymbols
{xr |i≥1, r ∈R}; the symbol xr indicates the function R→R defined by
i i
xr if x >0
x 7→ i i
i
(0 if xi ≤0.
He considered the structure
RRan := R,<,+,−, ·,0,1,(xri)r∈R,i≥1, f˜ f∈R{X,n},n∈N ,
where f˜ (cid:0) denotes a certain class of fu(cid:0)nc(cid:1)tions f˜: Rn(cid:1)→ R that are
f∈R{X,n},n∈N
analyticon[−1,1]n. HeprovedthatthetheoryofRR admitsquantifier-elimination
(cid:0) (cid:1) an
and analytic cell-decomposition, and is universally axiomatizable, o-minimal, and
polynomially bounded.
The standard properties of o-minimal theories (cf., e.g., [Dries, 1998] or [Miller,
1994])imply that the zeros in R2 ofall the various a∈B consistof finitely many
++
isolated points together with the graphs of finitely many continuous, monotonic
functionsξ :(γ ,γ )→R (onsuitableintervals(γ ,γ )⊆R )satisfy-
a,p,j p p+1 ++ p p+1 ++
ing(4.3.1),asstatedinthelemma. (Thattheξ aregeneralizedsemialgebraicis
a,p,j
7Wedonotneedthemonotonicity oftheξa,p,j inthispaper.
8We say that a function is generalized semialgebraic if its graph, in the product space, is a
generalizedsemialgebraicset.
9Here, K isas in(4.0.2); infact, sis even bounded by the number of alternations in sign in
the sequence a0(x),...,aK(x), by Sturm’s generalization [[cite]cite.Sturm 1829Sturm, 1829], to
one-variable generalized polynomials, of the Fourier-Budan theorem (which contains Descartes’
ruleofsignsasaspecialcase).
TWO-VARIABLE PIERCE-BIRKHOFF FOR GENERALIZED POLYNOMIALS 7
y 6s(0)=0 s(1) s(2)=3 s(3)=5 s(4) s(5)=3
=0 =5
D
2,3 ξ (=ξ3,4) ξc,4,2 ξc,5,2 (=ξ5,2)
b,3,2
ξ
b,2,2 ξ
(=ξ2,2) (cid:0)(cid:9) b,4,2
(=ξ4,3)
ξ
c,3,2 (=ξ3,3)
D
2,2
ξ (=ξ3,2)
c,3,1
(=ξ2,1) @I
ξb,2,1 ξb,4,1
ξ (=ξ3,1) (=ξ4,2)
b,3,1 ξ
c,4,1
r D2,1 ξc,5,1 (=ξ5,1)
AK
H H H H H H
0 A 1 2 3 4 5
a(x,y)=0
-
0 γ γ γ γ γ x
1 2 3 4 5
Figure 1. Illustrating Lemma 4.3 and Corollary 4.5 by showing
the zerosinR2 ofa,b,c∈B: theisolatedzeroofa(x,y), andthe
++
graphsofy =ξ (x) andy =ξ (x) (whicharealsothe graphs
b,p,j c,p,j
of y =ξp,k(x), for suitable k). Here, L=5 (the number of γ’s).
justthe definition ofthatterm(footnote 8 above),since the a(x,y) aregeneralized
polynomials.) (cid:3)
Notation 4.4. Itwillbe helpful in(4.5.1)belowif weagreethatξ (x)=0 and
a,p,0
ξ (x) = +∞ for all x ∈ (γ ,γ ), where p ∈ {0,1,...,L} and s = s(a,p) is
a,p,s+1 p p+1
as in Lemma 4.3.
Corollary 4.5. Let L, γ ,...,γ , and H be as in (4.3), for some fixed p ∈
0 L+1 p
{0,1,...,L}. Then the zeros in H of all the a ∈ B are the graphs of continu-
p
ous, monotonic, generalized semialgebraic functions y =ξp,k(x), k =1,2,...,s(p),
where s(p) satisfies 0 ≤ s(p) ≤ s(a,p) (where s(a,p) is as in (4.3)), and
a∈B
where, for each x∈(γ ,γ ),
p p+1
P
0=:ξp,0(x)<ξp,1(x)<···<ξp,s(p)(x)<ξp,s(p)+1(x):=∞. (4.5.1)
Consequently, the sets
D :={(x,y)|γ <x<γ , ξp,k(x)<y <ξp,k+1(x)},
p,k p p+1
for k ∈ {0,1,...,s(p)}, are nonempty, pairwise-disjoint, generalized semialgebraic
cells (in particular, they are open and (pathwise) connected), and their union is
a dense open subset of H . Moreover, the D are “stacked” one upon the other
p p,k
in the y-direction, so that for any x ∈ (γ ,γ ) and for any (s(p) + 1)-tuple
p p+1
y ,y ,...,y ∈R for which each (x,y )∈D , y <y <···<y .
0 1 s(p) ++ k p,k 0 1 s(p)
Proof. The required sequence ξp,1,ξp,2,...,ξp,s(p) of functions is just a suitable
permutationandrelabellingofthesetoffunctions{ξ |a∈B,1≤j ≤s(a,p)}.
a,p,j
That a permutation of the ξ’s satisfying (4.5.1) exists follows from (4.3.1). (cid:3)
8 CHARLESN.DELZELL
Proposition 4.6. The set of suprema of infima of finitely many generalized poly-
nomial functions is closed under subtraction and multiplication, and so is a ring.
Proof. This is aspecialcaseofaresultofHenriksenandIsbell[[cite]cite.Henriksen
et al. 1962Henriksen, et al., 1962, Corollary 3.4]: If S is a ring of real-valued
functions on a set, then the least lattice of functions that contains S is also a ring.
Here we may take S = R[R2] (1.1). For the proof of this corollary, Henriksen and
Isbell gavesome f-ring identities which, they said, reduce the proof to an exercise;
theyomittedthedetails. [Delzell,1989]gaveasketchofaproof. Thefirstcomplete
proofofthisfacttoappearinprintwasthatof[Hager,etal.,2010,Theorem1(B)];
their proof incorporates some simplifications due to Madden, and their statement
is a little more general than the Henriksen-Isbell statement above, in that now S
may be an arbitrary subring of an arbitrary f-ring. (cid:3)
Inthenextlemmaitwillhelpfultousethe abbreviationa+ =sup{0,a},forany
real-valued function a.
Lemma 4.7 (Generalized Mah´e lemma). Using the notation of Lemma 4.3 above,
for each p ∈ {0,1,...,L}, each a(x,y) ∈ B, and each j ∈ {0,1,...,s} (where
s = s(a,p) as in (4.3)), there exists a function c (x,y) that is a sup of infs of
a,p,j
finitely many generalized polynomials, such that for all x ∈ (γ ,γ ) and for all
p p+1
y ∈R ,
++
a(x,y) if y >ξ (x), and
a,p,j
c (x,y)= (4.7.1)
a,p,j
(0 otherwise.
Proof. Fix any p≤L.
We use induction on K ≥ 1, the number of distinct y-exponents occurring in a
(recall(4.0.2)). NotethatforanyK ≥1,wemay(infact,wemust)takec =a;
a,p,0
this handles the case K = 1, i.e., the case where a(x,y) is of the form a (x)yβ1
1
(which implies s(a,p)=0 for each p≤L).
Now assume K >1.
We claim that we may assume
β =0. (4.7.2)
1
If not, then write b(x,y) = y−β1a(x,y). Thus b ∈ B, by (4.0.3). Note that b(x,y)
has the same positive y-roots ξ as a(x,y) has; thus s(a,p) = s(b,p). Therefore, if
for each j ≤s(b,p) we can construct c such that
b,p,j
b(x,y) if y >ξ (x), and
b,p,j
c (x,y)=
b,p,j
(0 otherwise,
then we may, for each j ≤ s(a,p) (= s(b,p)), take c (x,y) = yβ1c (x,y); the
a,p,j b,p,j
latter product is a sup of infs of finitely many generalized polynomials, since c
b,p,j
is, and since yβ1 >0 for all y >0 (or use (4.6)).
Next,recallthata′(4.0.3)andr (4.0.4)eachhaveexactlyK−1y-terms,by(4.2)
a
and(4.7.2). Thusweassume,bytheinductivehypothesis,thatforeveryk≤s(a′,p)
and l ≤ s(ra,p), we can construct ca′,p,k and cra,p,l satisfying the appropriate
analogsof (4.7.1). Notethatca′,p,k andcra,p,l are,inparticular,continuous(either
by their formasin (4.7.1), orby the factthat they aresups of infs offinitely many
generalized polynomial functions).
TWO-VARIABLE PIERCE-BIRKHOFF FOR GENERALIZED POLYNOMIALS 9
Finally, in order to construct c , we now use induction on j ∈ {0,1,2,...,
a,p,j
s(a,p)}. We have already constructed c , so now we assume that j ∈{1,2,...,
a,p,0
s(a,p)} and that c has already been constructed with the properties stated
a,p,j−1
in Lemma 4.7.
Throughout the rest of this proof, x will range over (γ ,γ ). By the uni-
p p+1
form trichotomy in (4.3.1), all order relations involving the various ξ’s below will
hold uniformly for such x; thus we usually write, e.g., ξ instead of ξ (x).
a,p,j a,p,j
Let k be the smallest index such that ξa,p,j ≤ ξa′,p,k (then 1 ≤ k ≤ 1+s(a′,p)).
Let l be the smallest index such that ξa′,p,k ≤ ξra,p,l (then 1 ≤ l ≤ 1+s(ra,p)).
Then
ξa′,p,k <ξa,p,j+1 (unless ξa′,p,k =∞), by Rolle’s theorem, and (4.7.3)
y
g(x,y):= β ca′,p,k(x,y)+cra,p,l(x,y)
K
0 if 0<y <ξa′,p,k,
y
=β a′(x,y)=a(x,y)−ra(x,y) if ξa′,p,k <y <ξra,p,l, (4.7.4)
 yKa′(x,y)+r (x,y)=a(x,y) if ξ <y,
β a ra,p,l
K
where (4.7.4) follows from (4.0.4) and from the definitions of ca′,p,k and cra,p,l.10
This function g is a supremum of infima of finitely many generalized polynomial
functions, by (4.6).
If a′(x,ξ )=0, then
a,p,j
ξa′,p,k =ξa,p,j by the minimality of k, and
ξra,p,l =ξa′,p,k by (4.0.4) and the minimality of l.
Thus we may take c =g, by (4.7.4).
a,p,j
Now suppose, on the other hand, that
a′(x,ξ )6=0 (4.7.5)
a,p,j
(recall (4.3.1)). (Then
ξa,p,j <ξa′,p,k.) (4.7.6)
We may assume that in fact
a′(x,ξ )>0, (4.7.7)
a,p,j
by (4.3.1), by replacing a with −a, and by the fact that −c (= c ) will
−a,p,j a,p,j
still be a supremum of infima of finitely many generalized polynomial functions if
c is, by (4.6). Then
−a,p,j
a(x,y)<0 for ξ <y <a and (4.7.8)
a,p,j−1 a,p,j
a(x,y)>0 for ξ <y <a , (4.7.9)
a,p,j a,p,j+1
by (4.7.7).
10In (4.7.4), the inequalities in the case-distinctions y < ξa′,p,k, ξa′,p,k < y < ξra,p,l, and
ξra,p,l<yareallstrict(i.e.,theyareall<,andnot≤). Thisstrictnessisnecessarybecauseξa′,p,k
and/or ξra,p,l could be ∞. If either or both of the ξ’s are finite, the corresponding inequalities
could be relaxed to nonstrict inequalities (with ≤). But even without such a relaxation, (4.7.4)
stilluniquelydetermines g evenwheny isξa′,p,k orξra,p,l,sinceg iscontinuous forally>0.
10 CHARLESN.DELZELL
First suppose ξa′,p,k = ∞ (i.e., k = 1 + s(a′,p)). Then a′(x,y) > 0 for all
y > ξ , whence a(x,y) > 0 for all y > ξ . Hence we may take c =
a,p,j a,p,j a,p,j
inf{c+ ,a+}, using also (4.7.8).
a,p,j−1
Second, suppose ξa′,p,k <∞ (i.e., k ≤s(a′,p)). Then
ra(x,ξa′,p,k)=a(x,ξa′,p,k)− ξa′,p,ka′(x,ξa′,p,k) (by (4.0.4))
β
K
ξa′,p,k
=a(x,ξa′,p,k)− ·0
β
K
=a(x,ξa′,p,k)>0, by (4.7.9), (4.7.3), and (4.7.6). (4.7.10)
Then for ξa′,p,k ≤y <ξra,p,l:
r (x,y)>0 by (4.7.10) and the choice of l, and (4.7.11)
a
g(x,y)=a(x,y)−r (x,y) by (4.7.4)
a
<a(x,y) by (4.7.11). (4.7.12)
Then
a+ if 0<y ≤ξ by (4.7.4), and
a,p,j
sup{a,g}=
(a if y ≥ξa,p,j by (4.7.4), (4.7.12), (4.7.3), and (4.7.9).
Therefore, we may take c =inf{c+ , sup{a,g}}, by (4.7.8). (cid:3)
a,p,j a,p,j−1
Proposition 4.8. Let h, A, and B be as before Lemma 4.2, and let L and H be
p
as in Lemma 4.3, for some fixed p ∈ {0,1,...,L}. Then there is a function d :
p
R2 →R that (1) is a supremum of infima of finitely many generalized polynomial
++
functions ∈R[R2] and (2) coincides with h(x,y) on H .
p
Proof. Let γ and γ be as in Lemma 4.3, and let s(p), ξp,0,...,ξp,s(p)+1, and
p p+1
D ,...,D be as in Corollary 4.5.
p,0 p,s(p)
For each k =0,1,...,s(p) there exists a unique µ :=µ(p,k)∈{1,2,...,l} such
that D ⊆ A (hence h = g on D , by (4.0.1)), using Lemma 2.3(5) and the
p,k µ µ p,k
fact that each g −g is nonzero throughout D .
i j p,k
Ifs(p)=0,wemaydefinetherequiredd tobeg ∈R[R2]. Ifs(p)>0,then
p µ(p,0)
weshalldefined asfollows. Fork =0,1,...,s(p)−1,letv :=g −g .
p p,k µ(p,k+1) µ(p,k)
We have v =0 on D ∩D , since h is continuous. We extend the notation
p,k p,k p,k+1
c of Lemma 4.7 from the case where a ∈ B to the case where a = 0: for
a,p,j
j = 0,1,..., we define the function c by c (x,y) = 0 ∀(x,y) ∈ R2 . If
0,p,j 0,p,j ++
v 6= 0, then v ∈ A ⊂ B, so by (4.3) and (4.5) there exists a unique j(p,k) ∈
p,k p,k
{1,2,...,s(v ,p)} such that the graph of y = ξ (x) over (γ ,γ ) separates
k vk,p,j p p+1
D from D . We may now take
p,k p,k+1
s(p)−1
d =g + c(v ,p,j(p,k)),
p µ(p,0) p,k
k=0
X
by (4.7) and (4.6). (cid:3)
Remark 4.9. Theabovepropositionprovestheone-variableanalogofTheorem3.1.
For if the given function h does not involve one of the two variables (say, x), then
by Remark 4.1 above, none of the functions that we constructed in the sets A and
B willinvolvex,either,whencewewouldbeabletotakeL=0(whichwouldmean
that H equals all of R2 ) in (4.3)–(4.5), (4.7), and (4.8) above.
0 ++