#### Metric Properties of Semialgebraic Mappings

Discrete Comput Geom
Metric Properties of Semialgebraic Mappings
Krzysztof Kurdyka 0 1
Stanisław Spodzieja 0 1
Anna Szlachci n´ska 0 1
Krzysztof Kurdyka 0 1
Editor in Charge: Günter M. Ziegler
0 Faculty of Mathematics and Computer Science, University of Łódz ́ , S. Banacha 22, 90-238 Lodz , Poland
1 Laboratoire de Mathematiques (LAMA), Universié Savoie Mont Blanc, UMR-5127 de CNRS , 73-376, Le Bourget-du-Lac Cedex , France
We give various quantitative versions of Łojasiewicz inequalities for semialgebraic sets and mappings, both in the local and global case.
Łojasiewicz exponent; Semialgebraic set; Semialgebraic mapping; Polynomial mapping
1 Introduction
Łojasiewicz inequalities emerged in the late 1950s as the main tool in the division
of distributions by a real polynomial (Hörmander [
17
]) and by a real analytic
function (Łojasiewicz [25, 26]). Since then they have turned out to be of use in numerous
branches of mathematics, including differential equations, dynamical systems and
singularity theory (see for instance [24,28,38]). Quantitative versions of these
inequalities, involving e.g. computing or estimating the relevant exponents, are of importance
in real and complex algebraic geometry (see [43] and also [31–33]). Recently a strong
demand for explicit estimates of the Łojasiewicz exponent comes from optimization
theory (see for instance [23,37]) and also from estimates for global error bounds [27].
Our goal is to give various quantitative versions of these inequalities in the real case
both in the local and global context. We denote by K the field R of real numbers or
the field C of complex numbers.
Let X ⊂ KN be a closed semialgebraic set. (If K = C we consider X as a subset
of R2N .) Assume that 0 ∈ X is an accumulation point of X and f, g : X → R are
two continuous semialgebraic functions such that f −1(0) ⊂ g−1(0). Then there are
positive constants C, η, ε such that the following Łojasiewicz inequality holds (see
e.g. [
4
]):
| f (x )| ≥ C |g(x )|η if x ∈ X, |x | < ε.
(1.1)
The infimum of the exponents η in (1.1) is called the Łojasiewicz exponent of the
pair ( f, g) on the set X at 0 and is denoted by L0( f, g|X ). It is known (see [
3
])
that L0( f, g|X ) is a rational number; moreover, inequality (1.1) holds actually with
η = L0( f, g|X ) for some ε, C > 0 (see [41]). An asymptotic estimate for L0( f, g|X )
was obtained by Solernó [39]; we shall discuss it in Remark 2.4. Inequality (1.1) is
valid in a more general setting of functions definable in an o-minimal polynomially
bounded structure (in particular for subanalytic functions) (see [
12,16
]).
From the point of view of applications the most interesting case of inequality (1.1) is
when f is a semialgebraic function and g(x ) = dist(x , X ∩ f −1(0)). We shall consider
the distance induced by the Euclidean norm. By convention dist(x , ∅) = 1. More
precisely, we shall consider the following case. Let F = ( f1, . . . , fm ) : KN → Km
be a semialgebraic mapping and X ⊂ KN a closed semialgebraic set such that 0 ∈ X
is an accumulation point of X . So we have the following Łojasiewicz inequality:
|F (x )| ≥ C dist(x , F −1(0) ∩ X )η if x ∈ X, |x | < ε.
(1.2)
The smallest exponent η in (1.2) is called the Łojasiewicz exponent of F on the set X at
K K
0 and is denoted by L0 (F |X ). In Sect. 2 we shall give explicit bounds for L0 (F |X ) in
terms of the degrees of the data involved. The main result of this section is an explicit
estimate for the local Łojasiewicz exponent for separation of semialgebraic sets (see
Theorem 1.1).
The second aim of this article is to obtain similar results but for the Łojasiewicz
exponent at infinity. Assume now that a closed semialgebraic set X ⊂ KN is
unbounded. By the Łojasiewicz exponent at infinity of a mapping F : X → Km
we mean the supremum of the exponents ν in the following Łojasiewicz inequality:
|F (x )| ≥ C |x |ν for x ∈ X, |x | ≥ R,
for some positive constants C , R; we denote it by LK∞(F |X ). If X = KN we call
K
the exponent L∞(F |X ) the Łojasiewicz exponent at infinity of F and denote it by
(1.3)
LK∞(F ). Clearly LK∞(F |X ) may be negative. Note that inequality (1.3) holds only
when X ∩ F −1(0) is compact.
The next inequality, called the Hörmander–Łojasiewicz inequality [
17
], is always
valid for a continuous semialgebraic mapping:
|F (x )| ≥ C
dist(x , F −1(0) ∩ X ) θ
where C, θ are some positive constants. In Sect. 3 we state Theorem 3.2 which is a
global quantitative version of regular separation at infinity of semialgebraic sets. It
implies, in particular, an estimate for the exponent (see Corollaries 3.3, 3.4).
The paper is organized as follows: in Sects. 2 and 3 we discuss Łojasiewicz inequalities respectively in the local and global case. The proofs of the main results are given in the last section.
2 The Łojasiewicz Exponent at a Point
We will give an estimate from above of the Łojasiewicz exponent for the regular
separation of closed semialgebraic sets and for a continuous semialgebraic mapping
on a closed semialgebraic set. Let us start from some notation. Let X ⊂ RN be a
closed semialgebraic set. It is known that X has a decomposition
X = X1 ∪ · · · ∪ Xk
into the union of closed basic semialgebraic sets
Xi = {x ∈ RN : gi,1(x ) ≥ 0, . . . , gi,ri (x ) ≥ 0, hi,1(x ) = · · · = hi,li (x ) = 0},
i = 1, . . . k (see [
4
]), where gi,1, . . . , gi,ri , hi,1, . . . , hi,li ∈ R[x1, . . . , xN ]. Assume
that ri is the smallest possible number of the inequalities gi, j (x ) ≥ 0 in the definition
of Xi , for i = 1, . . . , k. Denote by r (X ) the minimum of max{r1, . . . , rk } over all
decompositions (2.1) into unions of sets of the form (2.2). As shown by Bröcker [
6
]
(cf. [
5,35
]),
(2.1)
(2.2)
(2.3)
r (X ) ≤
N (N + 1)
2
Denote by κ(X ) the mimimum of the numbers
max{deg g1,1, . . . , deg gk,rk , deg h1,1, . . . , deg hk,lk }
over all decompositions (2.1) of X into the union of sets of the form (2.2), provided
ri ≤ r (X ). Obviously r (X ) = 0 if and only if X is an algebraic set. The numbers
r (X ) and κ(X ) characterize the complexity of the semialgebraic set X . For more
information about the complexity see for example [
2,4,34
].
Theorem 2.1 Let X, Y ⊂ RN be closed semialgebraic sets, and suppose 0 ∈ X ∩ Y .
Set r = r (X ) + r (Y ) and d = max{κ(X ), κ(Y )}. Then there exist a neighbourhood
U ⊂ RN of 0 and a positive constant C such that
dist(x , X ) + dist(x , Y ) ≥ C dist(x , X ∩ Y )d(6d−3)N+r−1 for x ∈ U.
(2.4)
If, additionally, 0 is an isolated point of X ∩ Y , then for some neighbourhood U ⊂ RN
of 0 and some positive constant C ,
dist(x , X ) + dist(x , Y ) ≥ C |x |
(2d−1)2N+r +1
for x ∈ U.
(2.5)
The proof of the above theorem will be carried out in Sect. 4. The key point in
the proof will be the following inequality [22, Cor. 8]. Let X = (g1, . . . , gk )−1(0)
and Y = (h1, . . . , hl )−1(0) RN , where gi , h j ∈ R[x1, . . . , xN ] are polynomials of
degree not greater than d. Let a ∈ RN . Then there exists a positive constant C such
that
dist(x , X ) + dist(x , Y ) ≥ C dist(x , X ∩ Y )d(6d−3)N−1
in a neighbourhood of a. If, additionally, a is an isolated point of X ∩ Y , then
dist(x , X ) + dist(x , Y ) ≥ C |x − a|
(2d−12)N +1
in a neighbourhood of a for some positive C > 0, which is a consequence of [
14
].
Theorem 2.1 implies
Corollary 2.2 Let F : X → Rm be a continuous semialgebraic mapping, where
X ⊂ RN is a closed semialgebraic set, and suppose 0 ∈ X and F (0) = 0. Set
r = r (X ) + r ( graph F ) and d = max{κ(X ), κ( graph F )}. Then
If, additionally, 0 is an isolated zero of F , then
L0R(F |X ) ≤ d(6d − 3)N +r−1.
L0R(F |X ) ≤
(2d − 1)N +r + 1
2
Remark 2.3 The inequality (2.6) is crucial for estimating the rate of convergence of
algorithms (based on semi-definite programming) of minimization of a polynomial
on a basic semialgebraic set. Indeed, (2.6) enabled us [23] to reduce effectively the
problem of minimizing polynomials on a compact semialgebraic set to the case of
minimizing polynomials on a ball, which is much simpler [36].
Remark 2.4 We shall now comment on the result of Solernó [39] concerning the
Łojasiewicz exponent L0( f, g|X ) in the inequality (1.1) for a pair ( f, g) of continuous
(KS1)
(G)
(2.6)
(2.7)
semialgebraic functions on a closed semialgebraic set X ⊂ RN . In general his estimate
is of the form
L0( f, g|X ) ≤ D Mca ,
where D is a bound for the degrees of the polynomials involved in a description of f ,
g and X ; M is the number of variables in these formulas (so in general M ≥ N ); a
is the maximum number of alternating blocs of quantifiers in these formulas; and c is
an (unspecified) universal constant. The estimate (Sa ) was obtained from the effective
Tarski–Seidenberg theorem [15].
In our Corollary 2.2 only the function g(x ) = dist(x , X ∩ F −1(0)) is defined by
a formula which is not quantifier-free, and it has two alternating blocs of quantifiers,
hence a = 2. So Solernó’s estimate (Sa ) reads L0R(F |X ) ≤ d(N +2)2c , which is
comparable with our estimate L0R(F |X ) ≤ d(6d − 3)N +r−1 since r (X ) ≤ 21 N (N + 1) by
(2.3). Indeed, we believe that the universal constant c is at least 1, probably c 1.
Needless to say, our estimate is explicit.
Recall that for a real polynomial mapping F : RN → Rm such that d = deg F
(where deg F is the maximum of the degrees of the components of F ) we have
L0R(F ) ≤ d(6d − 3)N −1
(see [22, Cor. 6] or [29]). Actually both papers are based on an estimate for the
Łojasiewicz exponent in the gradient inequality obtained in [
11,13
].
We now consider a polynomial mapping restricted to an algebraic set. From
Corollary 2.2 we obtain an estimation of its local Łojasiewicz exponent, also for a non-isolated zero-set (cf. [30,40] for mappings with isolated zeros).
Corollary 2.5 Let F : (KN , 0) → (Km , 0) be a polynomial mapping, let X ⊂ KN
be an algebraic set defined by a system of equations g1(x ) = · · · = gr (x ) = 0, where
g1, . . . , gr ∈ K[x1, . . . , xN ], and let d = max{deg F, deg g1, . . . , deg gr }. Assume
that d > 0 and 0 ∈ X .
(a) If K = R, then L0R(F |X ) ≤ d(6d − 3)N −1.
(b) If K = C, then L0C(F |X ) ≤ d N .
Indeed, assertion (a) immediately follows from Corollary 2.2. We will prove (b).
Let G = (F, g1, . . . , gr ) : CN → Cm+r . We can assume that m ≥ N . Similarly to
[C4m2,oTfhthme. f1o]r,mweLpir(oyv1e, .th. a.t,tyhmer)e=exyisit+sa linmje=arr+m1aαpi,pjiynjg, iL==1(,L.1.,. .,.m. ,, wLhme)re: Cαim,j+∈r →C,
such that L0C(G|X ) = L0C(L ◦ G|X ). Moreover, deg L j ◦ G ≤ d for j = 1, . . . , m.
Cygan [
8
] proved that for analytic sets Z , Y ⊂ CN +m the intersection index at 0 of
Z and Y is a separation exponent of Z and Y at 0 ∈ Z ∩ Y . It is known that for
Z = CN × {0} and Y = graph L ◦ G, the index does not exceed d N (see [
10,44
]),
so L0C(L ◦ G) ≤ d N . Since G−1(0) = F −1(0) ∩ X and by definition of L we have
G(x ) = (F (x ), 0) for x ∈ X , it follows that L0C(F |X ) ≤ d N , proving (b).
(Sa )
(KS2)
Cha˛dzyn´ski [7] proved that
C
L∞(F ) ≥ d2 − d1d2 +
b∈F−1(0)
μb(F ),
where μb(F ) is the multiplicity of F at b, provided N = m = 2 and # F −1(0) < ∞.
For arbitrary m ≥ N , under the assumption # F −1(0) < ∞, Kollár [21] proved that
C
L∞(F ) ≥ dm − B(d1, . . . , dm ; N );
then Cygan et al. [
10
] improved this to
C
L∞(F ) ≥ dm − B(d1, . . . , dm ; N ) +
μb(F ),
(CKT)
where μb(F ) is the intersection multiplicity (in the sense of Achilles et al. [
1
]) of the
graph of F and Cn × {0} at the point (b, 0). For a complex k-dimensional algebraic
variety V ⊂ CN of degree D the following estimate was obtained by Jelonek [
18,19
]:
C
L∞(F |V ) ≥ dm − D · B(d1, . . . , dm ; k) +
μb(F ),
(J)
b∈F−1(0)
b∈F−1(0)∩V
3 The Łojasiewicz Exponent at Infinity
Let us first recall some known results on the Łojasiewicz exponent at infinity of a
pdeofilynniotimoniaolfmLaC∞pp(iFng|VF) =and( fL1,C∞.(. F.,) fimn )In:tCroNdu→ctioCnm). oLnetandeaglgfejbr=aicd sje,tjV=⊂1,C. N.. (,smee,
d1 ≥ · · · ≥ dm > 0 and set
B(d1, . . . , dm ; k) =
d1 · · · dm
d1 · · · dk−1dm
for
for
m ≤ k,
m > k.
(Ch)
(K)
(C1)
(C2)
where #(F −1(0) ∩ V ) < ∞. Cygan [
9
] gave the following global inequality:
|F (x )| ≥ C
for some positive constant C . Moreover she proved in [
8
] that for complex algebraic
sets X, Y ⊂ CN there exists a positive constant C such that
dist(x , X ) + dist(x , Y ) ≥ C
A result similar to (C2) was obtained by Ji et al. [20].
For real algebraic sets we have the following global Łojasiewicz inequality (see
[22]). If X, Y ⊂ RN are algebraic sets defined by systems of polynomial equations of
degrees at most d, then for some positive constant C ,
In particular, we have the following global Łojasiewicz inequality (see [22]). Let
F = ( f1, . . . , fm ) : RN → Rm be a polynomial mapping of degree d. Then for some
positive constant C ,
Moreover, if the set F −1(0) is compact, then
R
L∞(F ) ≥ −d(6d − 3)N −1.
(KS4)
(KS5)
By using (KS4) we obtain a global Łojasiewicz inequality for polynomial mappings.
Proposition 3.1 Let X ⊂ RN be an algebraic set defined by a system of polynomial
equations g1(x ) = · · · = gr (x ) = 0, where g1, . . . , gr ∈ R[x1, . . . , xN ]. Let F :
RN → Rm be a polynomial mapping and let d = max{deg F, deg g1, . . . , deg gr }.
Then for some positive constant C ,
Indeed, let G = (g1, . . . , gr ) : RN → Rr , and let H : RN → Rm+r be a
polynomial mapping defined by H (x ) = (F (x ), G(x )) for x ∈ RN . Then H −1(0) =
F −1(0) ∩ X , so from (KS4) we deduce the first assertion. If F −1(0) ∩ X is compact,
then so is H −1(0), and the second assertion follows immediately from the first (cf.
(KS5)).
In the above proof we cannot apply (Ch), (K), (CKT), (J) or (C1), because the
complexification of a real polynomial mapping with compact real zero-set may have
an unbounded zero-set.
The following global Łojasiewicz inequality for semialgebraic sets is the main result of this section. The proof is given in Sect. 4.
Theorem 3.2 Let X, Y ⊂ RN be closed semialgebraic sets. Set r = r (X ) + r (Y ) and
d = max{κ(X ), κ(Y )}. Then there exists a positive constant C such that
dist(x , X ) + dist(x , Y ) ≥ C
Theorem 3.2 immediately implies the following.
Corollary 3.3 Let F : X → Rm be a continuous semialgebraic mapping, where
X ⊂ RN is a closed semialgebraic set. If d = max{κ(X ), κ(Y )} and r = r (X )+r (Y ),
where Y = graph F , then there exists a positive constant C such that
In particular, if the set X is unbounded and F −1(0) ∩ X is compact, then
R
L∞(F |X ) ≥ (1 − d)d(6d − 3)N +r−1.
For a polynomial mapping F : X → Rm we have r ( graph F ) = r (X ) and
κ( graph F ) = max{deg F, κ(X )}, so we obtain
Corollary 3.4 Let F : X → Rm be a polynomial mapping, where X ⊂ RN is a closed
semialgebraic set. If D = max{2, κ(X )} and d = max{deg F, D}, and r = 2r (X ),
then
In particular, if the set X is unbounded and F −1(0) ∩ X is compact, then
LR∞(F |X ) ≥ − D2 d(6d − 3)N +r−1.
The above corollary is not a direct consequence of Corollary 3.3, so we will prove it separately in Sect. 4.
4 Proofs of Theorems 2.1 and 3.2 and of Corollary 3.4
It suffices to consider the case when X and Y are basic closed semialgebraic sets. So, let
X = {x ∈ RN : g1,1(x ) ≥ 0, . . . , g1,r(X)(x ) ≥ 0, h1,1(x ) = · · · = h1,l (x ) = 0},
Y = {x ∈ RN : g2,1(x ) ≥ 0, . . . , g2,r(Y )(x ) ≥ 0, h2,1(x ) = · · · = h2,l (x ) = 0},
where gi, j , hi,s ∈ R[x1, . . . , xN ]. We may indeed assume that X and Y are defined by
the same number of equations, because we can repeat the same equations if necessary.
Let r1 = r (X ), r2 = r (Y ), r = r1 + r2, and let Gi : RN × Rr → Rri , i = 1, 2, be the
polynomial mappings defined by
G1(x , y1, . . . , yr ) = (g1,1(x ) − y12, . . . , g1,r1 (x ) − yr21 ),
G2(x , y1, . . . , yr ) = (g2,1(x ) − yr21+1, . . . , g2,r2 (x ) − yr21+r2 ).
∀x1∈X ∀x2∈Y ∃y∈Rr (x1, y) ∈ A ∧ (x2, y) ∈ B;
∀x∈RN \X ∃x1∈X ∀y∈Rr [dist(x , X ) = |x − x1| ∧ (x1, y) ∈ A ⇒
dist(x , X ) ≥ dist((x , y), A)]
(4.1)
(4.2)
(4.3)
Let
moreover,
and
∀x∈RN \Y ∃x2∈Y ∀y∈Rr [dist(x , Y ) = |x − x2| ∧ (x2, y) ∈ B ⇒
dist(x , Y ) ≥ dist((x , y), B)].
Indeed, we will prove (4.2); the proof of (4.3) is similar. Take x ∈ RN \X and let
x1 ∈ X satisfy dist(x , X ) = |x − x1|. So, for any y ∈ Rr such that (x1, y) ∈ A, we
have
dist(x , X ) = |x − x1| = |(x , y) − (x1, y)| ≥ dist((x , y), A).
This gives (4.2).
Proof of Theorem 2.1 We will assume that the origin is a non-isolated point of X ∩ Y ;
otherwise, we proceed in the same way using formula (G) instead of (KS1). Let
p = d(6d − 3)N +r−1.
Claim 1. The assertion (2.4) is equivalent to
dist(x , Y ) ≥ C dist(x , X ∩ Y ) p for x ∈ (∂ X ) ∩ U1
(4.4)
for a neighbourhood U1 = {x ∈ RN : |x | < ρ} of the origin, ρ < 1, and some
positive constant C , where ∂ X denotes the boundary of X (cf. [9, Lem. 4.2] and
[22, Proof of Theorem 2]). Indeed, the implication (2.4)⇒(4.4) is obvious. Assume
that the converse fails. Then for a neighbourhood U2 = {x ∈ RN : |x | < ρ2 } of the
origin, there exists a sequence aν ∈ U2 such that aν → 0 and
1
ν
Taking a subsequence if necessary, it suffices to consider two cases: aν ∈/ X for ν ∈ N
or aν ∈ Int X for ν ∈ N.
Assume that aν ∈/ X for ν ∈ N. Let xν ∈ (∂ X ) ∩ U1 be such that dist(aν , X ) =
1
|aν − xν |. Since ρ < 1, we have dist(aν , X ) p ≥ dist(aν , X ). So, for some C > 0,
1 1
[dist(aν , X ) + dist(aν , Y )] p ≥ dist(aν , X ) p ≥ C dist(aν , X ),
and, by (4.4),
1 1
[dist(aν , X ) + dist(aν , Y )] p ≥ dist(xν , Y ) p ≥ C dist(xν , X ∩ Y ).
Since dist(aν , X )+dist(xν , X ∩Y ) ≥ dist(aν , X ∩Y ), by adding the above inequalities,
we obtain
dist(aν , X ∩ Y ) p ≤ ν
This contradicts (4.4) and proves the claim in this case. Summing up, we have proved
Claim 1.
If d = 1, then the assertion is trivial. Assume that d > 1. By (KS1), there exists a
positive constant C such that
dist((x , y), A) + dist((x , y), B) ≥ C dist((x , y), A ∩ B)d(6d−3)N+r−1
in a neighbourhood W of 0 ∈ RN +r . Obviously, for any (x , y) ∈ RN +r ,
dist((x , y), A ∩ B) ≥ dist(x , X ∩ Y ).
(4.6)
(4.7)
and
One can assume that gi, j (0) = 0 for any i, j . Indeed, if gi, j (0) < 0 for some i, j ,
then 0 ∈/ X or 0 ∈/ Y , which contradicts the assumption. If gi, j (0) > 0 for some i, j ,
then we can omit this inequality in the definition of X , respectively Y , and the germ
at 0 of X , respectively Y will not change. If gi, j (0) > 0 for any i, j , then the assertion
reduces to (KS2). So, there exists a neighbourhood W1 = U3 × U × U ⊂ W of
0 ∈ RN +r , where U3 ⊂ RN , U ⊂ Rr(X) and U ⊂ Rr(Y ) such that:
for any (x1, y , y ) ∈ A, where x1 ∈ RN , y ∈ Rr(X), y ∈ Rr(Y )
for any (x2, y , y ) ∈ B, where x2 ∈ RN , y ∈ Rr(X), y ∈ Rr(Y )
if x1 ∈ X ∩ U3, then y ∈ U
if x2 ∈ Y ∩ U3, then y ∈ U .
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
Let U ⊂ U3 be a neighbourhood of 0 ∈ RN . If (∂ X ) ∩ U = ∅, then U ⊂ X and
the assertion is obvious. Assume that (∂ X ) ∩ U = ∅. Take x ∈ (∂ X ) ∩ U , and let
x ∈ Y be a point for which dist(x , Y ) = |x − x |. By (4.1) there exists y ∈ Rr such
that (x , y) ∈ A and (x , y) ∈ B. Diminishing the neighbourhood U if necessary, we
may assume that x ∈ U3. By (4.8) and (4.9) we see that (x , y) ∈ W , so, by (4.2) and
(4.3),
dist(x , Y ) ≥ dist((x , y), A) + dist((x , y), B).
Summing up, (4.6), (4.7) and Claim 1 give the assertion.
Proof of Theorem 3.2 Let p = d(6d − 3)N +r−1. If d = 1 then the assertion is trivial.
If X \Y = ∅ or Y \X = ∅, then the assertion is obvious. So, we will assume that
X \Y = ∅, Y \X = ∅ and d > 1. In particular ∂ X = ∅.
By (KS3) we have
for (x , y) ∈ RN +r .
Claim 2. The assertion (3.1) is equivalent to
dist(x , X ∩ Y ) p
for x ∈ ∂ X
dist((x , y), A) + dist((x , y), B) ≥ C
dist((x , y), A) + dist((x , y), B) ≥ C
dist((x , y), A ∩ B) p
for (x , y) ∈ RN +r . Since dist((x , y), A ∩ B) ≥ dist(x , X ∩ Y ) for any (x , y) ∈ RN +r
(see (4.7)), the inequality (4.10) gives
for some positive constant C (cf. [9, Lem. 4.2] and [22, Proof of Theorem 2]). Indeed,
the implication (3.1)⇒(4.12) is obvious. Assume that the converse fails. Then there
exists a sequence aν ∈ RN such that
and, by (4.12) and the fact that |aν | → ∞ and |bν − aν | → 0,
1 1
[dist(aν , X ) + dist(aν , Y )] p ≥ dist(bν , Y ) p ≥ C
dist(aν , X )
1 + |aν |d
,
dist(bν , X ∩ Y )
1 + |aν |d
.
Since dist(aν , X )+dist(bν , X ∩Y ) ≥ dist(aν , X ∩Y ), by adding the above inequalities
we obtain
.
This contradicts (4.13) and proves the claim in this case.
Consider now the case aν ∈ Int X for ν ∈ N. Let yν ∈ Y be such that dist(aν , Y ) =
|aν − yν |. By (4.13) we see that yν ∈/ X , so there exist xν ∈ (∂ X ) ∩ [aν , yν ] for ν ∈ N.
By (4.13), for sufficiently large ν,
This together with (4.13) gives
By (4.14), for sufficiently large ν we have |xν | ≤ 2|aν |, so, for a positive constant
C ,
dist(xν , Y ) ≤
C
ν
.
This contradicts (4.12) and proves the claim in this case. Summing up, we have proved
Claim 2.
Take any x0 ∈ ∂ X . By (4.1), (4.2) and (4.3) there exist x2 ∈ Y and y0 ∈ Rr such
that (x0, y0) ∈ A, (x2, y0) ∈ B, and dist(x0, Y ) = |x0 − x2| ≥ dist((x0, y0), B).
Hence from (4.11),
dist(x0, Y ) ≥ C
dist(x0, X ∩ Y ) p
It is easy to observe that there exist constants C1, R1 > 0 such that for (x , y) ∈ A,
|(x , y)| ≥ R1 we have C1|y|2 ≤ |x |d . Since d ≥ 2, for a constant C2 > 0 we obtain
|(x , y)| ≤ C2|x |d/2 for (x , y) ∈ A, |(x , y)| ≥ R1. Hence from (4.15) we easily
deduce
provided |x0| ≥ R1. So, diminishing C if necessary, we obtain (4.16) for all x0 ∈ ∂ X .
This together with Claim 2 gives the assertion of Theorem 3.3.
Proof of Corollary 3.4 Let H : RN +r → Rm+r+l be a polynomial mapping defined
by
H (x , y) = (F (x ), G1(x , y), h1,1(x ), . . . , h1,l (x )), x ∈ RN , y ∈ Rr .
Then deg H ≤ d. Let V = F −1(0) ∩ X and let Z = H −1(0). By (KS4), for some
positive constant C we have
|H (x , y)| ≥ C
dist((x , y), Z ) d(6d−3)N+r−1
for (x , y) ∈ RN × Rr .
Because dist((x , y), Z ) ≥ dist(x , V ), we obtain
|H (x , y)| ≥ C
dist(x , V )
It is easy to observe that there exist constants C1, R1 > 0 such that for (x , y) ∈ A
with |(x , y)| ≥ R1 we have C1|y|2 ≤ |x |D. Since D ≥ 2, for a constant C2 > 0
we obtain |(x , y)| ≤ C2|x |D/2 for (x , y) ∈ A, |(x , y)| ≥ R1. Hence from (4.17) we
easily deduce (3.2) for x ∈ X , |x | ≥ R1. So, diminishing C if necessary, we obtain
(3.2) for all x ∈ X .
(4.15)
(4.16)
We now show the second assertion of the corollary. Since X is unbounded, we
may assume that so is A. Since V is compact, so is H −1(0). By (KS5) we have
L∞( H ) ≥ −d(6d − 3)N +r −1, in particular for some constants C, R > 0,
| H (x , y)| ≥ C |(x , y)|−d(6d−3)N+r−1
for (x , y) ∈ A, |(x , y)| ≥ R.
Since |(x , y)| ≤ C2|x |D/2 for (x , y) ∈ A, |(x , y)| ≥ R1, it follows that, for some
constant C3 > 0.
|F (x )| = | H (x , y)| ≥ C3|x |− D2 d(6d−3)N+r−1
for (x , y) ∈ A, |(x , y)| ≥ R,
and LR∞(F | X ) ≥ − D2 d(6d − 3)N +r −1. This ends the proof of Corollary 3.4.
Acknowledgments This research was partially supported by OPUS Grant No. 2012/07/B/ST1/03293
(Poland) and ANR project STAAVF (France).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
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