¨ A FRANK-WOLFE TYPE THEOREM AND HOLDER-TYPE GLOBAL ERROR BOUNDS FOR GENERIC POLYNOMIAL SYSTEMS ‡ ˆ P D ` † , AND TIE ˆ´N SO.N PHA - INH† , HUY VUI HA S˜I TIE M Abstract This paper studies generic polynomial systems More precisely, let f0 and f1 , , fp : Rn → R be convenient polynomial functions, and let S := {x ∈ Rn | fi (x) ≤ 0, i = 1, , p} = ∅ The following results are shown: (i) A Frank-Wolfe type Theorem: Suppose that the map (f0 , f1 , , fp ) : Rn → Rp+1 is non-degenerate at infinity If f0 is bounded from below on S, then f0 attains its infimum on S; (ii) A H¨ older-type global error bound: Suppose that the map (f1 , , fp ) : Rn → Rp is non-degenerate at infinity Let d := maxi=1, ,p deg fi and H(d, n, p) := d(6d − 3)n+p−1 Then there exists a constant c > such that cd(x, S) ≤ [f (x)]+H(d,n,p) + [f (x)]+ for all x ∈ Rn , where d(x, S) denotes the Euclidean distance between x and the set S, f (x) := maxi=1, ,p fi (x) and [f (x)]+ := max{f (x), 0}; and (iii) For polynomial maps with fixed Newton polyhedra, the property of being nondegenerate at infinity is generic Introduction Let f0 and f1 , , fp : Rn → R be polynomial functions in the variable x ∈ Rn Let S := {x ∈ Rn | f1 (x) ≤ 0, , fp (x) ≤ 0}, and suppose throughout that S is nonempty Consider the following constrained optimization problem (1) inf f0 (x) such that x ∈ S The purpose of this paper is twofold Firstly, we are concerned with the question of existence of optimal solutions to the problem (1) In the case when all fi , i = 0, , p, Date: November 26, 2012 1991 Mathematics Subject Classification Primary 32B20; Secondary 14P, 49K40 Key words and phrases Error bounds, Frank-Wolfe type theorem, Newton polyhedron, nondegenerate polynomial maps † These authors were partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) grant 101.01-2011.44 ‡ This author’s research was partially supported by Vietnam National Foundation for Science and Tech- nology Development (NAFOSTED) grant 101.01-2010.08 are linear, it is well known that the set of optimal solutions is nonempty provided the problem is bounded below In 1956, Frank and Wolfe [20] proved that if fi ’s remain affine linear functions for i = 1, , p, and f0 is an arbitrary quadratic polynomial, then f0 being bounded from below over S implies that an optimal solution exists If the statement holds with respect to other classes of polynomial functions f0 , , fp we will speak of a Frank-Wolfe type theorem Many other authors generalized the Frank-Wolfe theorem to broader classes of functions For example, Perold [55] generalized the Frank-Wolfe theorem to a class of non-quadratic objective functions and linear constraints Andronov et al [2] extended the Frank-Wolfe theorem to the case of a cubic polynomial objective function f0 under linear constraints Luo and Zhang [43] also extended the Frank-Wolfe theorem to various classes of general convex/non-convex quadratic constraint systems More recently, Belousov and Klatte in [6] (see also [5]) showed that this result is still true if f0 , f1 , , fp are convex polynomials of arbitrary degree Secondly, we are interested in the question of whether one can use the residual (constraint violation) at a point x ∈ Rn to bound the distance from x to the set S More precisely, we study if there exist some positive constants c, α, and β such that (2) cd(x, S) ≤ [f (x)]α+ + [f (x)]β+ for all x ∈ Rn , where d(x, S) denotes the Euclidean distance between x and the set S, f (x) := maxi=1, ,p fi (x) and [f (x)]+ := max{f (x), 0} An expression of this kind is called a global error bound for the set S We say that a H¨older-type global error bound holds for the set S if the inequality (2) holds with the exponent β = The study of error bounds has grown significantly and has found many important applications For a summary of the theory and applications of error bounds, we refer the readers to the survey of Pang [54] and the references cited therein The first error bound result is due to Hoffman [27] His result deals with the case where the polynomials f1 , , fp are affine and states that the global error bound (2) holds with the exponents α = β = After the work of Hoffman, a lot of researchers have devoted themselves to the study of global error bound; see, for example, [3, 16, 30, 31, 35, 46, 51, 58] Under the convexity assumption of the polynomials fi , global H¨older-type error bounds have been shown in [36, 37, 38, 41, 42, 44, 45, 50, 61, 60] In the absence of convexity, global H¨older-type error bounds (even global error bounds) are highly unlikely to hold When the constrained set S defined by some affine linear functions and a single quadratic polynomial, Luo and Sturm [44] showed that the H¨older-type global error bound (2) holds with the exponents α = and β = In particular, a global error bound was obtained by H V H`a [26] for a nonlinear inequality defined by a single convenient polynomial, which is (Newton) non-degenerate at infinity (see [32] and Section for precise definitions) In this paper, we consider the class of polynomial maps which are (Newton) non-degenerate at infinity This notion extends the definitions of non-degenerate for analytic functions, in the (local and at infinity) complex setting [29, 32] It is worth paying attention to the fact that Non-degenerate at infinity polynomial maps have a number of remarkable properties which make them an attractive domain for various applications The main contributions of this paper are as follows: (i) Suppose that the map (f0 , f1 , , fp ) : Rn → Rp+1 is non-degenerate at infinity, and all fi are convenient, i = 0, 1, , p If the objective function f0 is bounded from below on the constrained set S, then f0 attains its infimum on S; (ii) Suppose that the map (f1 , , fp ) : Rn → Rp is non-degenerate at infinity, and all fi are convenient, i = 1, , p Then there exists a constant c > such that the following H¨older-type global error bound holds cd(x, S) ≤ [f (x)]+H(d,n,p) + [f (x)]+ for all x ∈ Rn , where d := maxi=1, ,p deg fi and H(d, n, p) := d(6d − 3)n+p−1 (iii) The class of polynomial maps (with fixed Newton polyhedra), which are non-degenerate at infinity, is generic in the sense that it is an open and dense semi-algebraic set It should be emphasized that we not require the polynomials fi to be convex, and their degrees can be arbitrary Moreover, our method is actually different from the argument in [26]: the proofs use only the Curve Selection Lemma (see Lemma 2.1) as a tool The results presented in the paper suggest that the class of polynomial maps, which nondegenerate at infinity, may offer an appropriate domain on which the machinery of polynomial optimization works with full efficiency The paper is structured as follows Section presents some backgrounds in the field In Section 3, we establish a Frank-Wolfe type theorem Some H¨older-type global error bound results will be given in Section Finally, in Section 5, we show that the property of being nondegenerate at infinity is generic Preliminaries In this section, we give the notations, definitions, and preliminary results which will used throughout the paper Throughout this paper, Rn denotes Euclidean space with dimension n The corresponding x, x inner product (resp., norm) in Rn is defined by x, y for any x, y ∈ Rn (resp., x := n for any x ∈ R ) 2.1 Semi-algebraic geometry In this subsection, we recall some notions and results of semi-algebraic geometry, which can be found in [4, 7, 8, 10, 18] Definition 2.1 of the form (i) A subset of Rn is called semi-algebraic if it is a finite union of sets {x ∈ Rn | fi (x) = 0, i = 1, , k; fi (x) > 0, i = k + 1, , p} where all fi are polynomials (ii) Let A ⊂ Rn and B ⊂ Rp be semi-algebraic sets A map F : A → B is said to be semi-algebraic if its graph {(x, y) ∈ A × B | y = F (x)} is a semi-algebraic subset in Rn × Rp Semi-algebraic sets and functions enjoy a number of remarkable properties: (i) The class of semi-algebraic sets is closed with respect to Boolean operators; a Cartesian product of semi-algebraic sets is a semi-algebraic set; (ii) The closure and the interior of a semi-algebraic set is a semi-algebraic set; (iii) A composition of semi-algebraic maps is a semi-algebraic map (iv) The image and inverse image of a semi-algebraic set under a semi-algebraic map are semi-algebraic sets (v) If S is a semi-algebraic set, then the distance function d(·, S) : Rn → R, x → d(x, S) := inf{ x − a | a ∈ S}, is also semi-algebraic A major fact concerning the class of semi-algebraic sets is its stability under linear projections (see, for example, [7, 10]) Theorem 2.1 (Tarski-Seidenberg Theorem) Let π(x1 , , xn ) = (x1 , , xn−1 ) be the canonical projection from Rn onto Rn−1 If S is a semi-algebraic subset of Rn , then so is π(S) in Rn−1 Remark 2.1 As an immediate consequence of Tarski-Seidenberg Theorem, we get semialgebraicity of any set {x ∈ A | ∃y ∈ B, (x, y) ∈ S}, provided that A, B, and S are semialgebraic sets in the corresponding spaces It follows that also {x ∈ A | ∀y ∈ B, (x, y) ∈ S} is a semi-algebraic set as its complement is the union of the complement of A and the set {x ∈ A | ∃y ∈ B, (x, y) ∈ S} Thus, if we have a finite collection of semi-algebraic sets, then any set obtained from them with the help of a finite chain of quantifiers is also semi-algebraic We will need a version of the Curve Selection Lemma Milnor [48] has proved this lemma at points of the closure of a semi-algebraic set N´emethi and Zaharia [49] showed how to extend the result at infinity at some fibre of a polynomial maps We give here a more general statement, and for the sake of completeness we include a proof of this fact Lemma 2.1 (Curve Selection Lemma at infinity) Let A ⊂ Rn be a semi-algebraic set, and let F := (f1 , , fp ) : Rn → Rp be a semi-algebraic map Assume that there exists a sequence xk ∈ A such that limk→∞ xk = ∞ and limk→∞ F (xk ) = y ∈ (R)p , where R := R ∪ {±∞} Then there exists an analytic curve ϕ : (0, ) → A of the form ϕ(t) = a0 tq + a1 tq+1 + · · · such that a0 ∈ Rn \ {0}, q < 0, q ∈ Z, and that limt→0 F (ϕ(t)) = y Proof Replacing if necessary fi by ±1 , there is no 1+(fi (x))2 n n+1 × Rp given by We consider the semi-algebraic map Φ : R → R Φ(x) := x1 1+ x , , xn 1+ x loss of generality to assume y ∈ Rp , 1+ x , F (x) It follows that we can suppose that the sequence Φ(xk ) is convergent to some point (u, y) ∈ Sn × Rp By Tarski-Seidenberg theorem, B := Φ(A) is a semi-algebraic set Thus we can apply the Curve Selection Lemma from [8, 10] for the point (u, y) ∈ B We obtain an analytic curve ψ(t) in B, which tends to (u, y) when t → +0 The desired curve ϕ(t) could be easily obtained from ψ(t) The following useful result is well-known (see, e.g., [18, 47]); for completeness we provide a short proof below Lemma 2.2 (Growth Dichotomy Lemma) Let f : (0, ) → R be a semi-algebraic function with f (t) = for all t ∈ (0, ) Then there exist constants c = and q ∈ Q such that f (t) = ctq + o(tq ) as t → 0+ Proof The set {(t, f (t)) ∈ R2 | < t < } is semi-algebraic By the Curve Selection Lemma [8, 10], there exist δ > and a parametrized analytic curve (x(s), y(s)), s ∈ (−δ, δ), such that x(0) = 0, x(s) > and f (x(s)) = y(s) for s ∈ (0, δ) By a change of the parameter s we can assume that x(s) = sk , for some positive integer k Then f (t) = y(t1/k ) has the desired form 2.2 The transversality theorem with parameters Let P, X and Y be some C ∞ manifolds of finite dimension Let S be a C ∞ sub-manifold of Y Let F : X → Y be a C ∞ map Denote dx F : Tx X → TF (x) Y , the derivative map of F at x, where Tx X and TF (x) Y are, respectively, the tangent space of X at x and Y at F (x) Definition 2.2 The map F is said to be transverse to the sub-manifold S, abbreviated F S, if either F (X) ∩ S = ∅ or we have for each x ∈ F −1 (S), dx F (Tx X) + TF (x) S = TF (x) Y Remark 2.2 If dim X ≥ dim Y and S = {s}, then F S if and only if either F −1 (s) = ∅ or rankdx F = dim Y for all x ∈ F −1 (s) Moreover, if dim X < dim Y , then F S if and −1 only if F (S) = ∅ The following result is useful in the sequel (see [23, 24]) Theorem 2.2 (Transversality Theorem) Let F : P ×X → Y be a C ∞ map For each p ∈ P, consider the map Fp : X → Y defined by Fp (x) := F (p, x) If F transversal to S, then the set D = {p ∈ P | Fp S} is open and dense in P Moreover, if P, X, Y, and S are semi-algebraic sets and if F is a semi-algebraic map, then D is also semi-algebraic Proof The proof of openness and density of D is done in [23, 24] The method used also permits to prove that D is semi-algebraic if P, X, Y, S and F are semi-algebraic 2.3 The set of asymptotic critical values Let F = (f1 , , fp ) : Rn → Rp be a C -map, and define the Rabier function νF : Rn → R by p νF (x) := λi ∇fi (x) p i=1 |λi |=1 i=1 Remark 2.3 (i) By definition, νF (x) = if and only if the gradient vectors ∇f1 (x), , ∇fp (x) are linearly dependent (ii) If the map F is semi-algebraic then so is νF Definition 2.3 [33, 57] We define the set of asymptotic critical values of F as ˜ ∞ (F ) := {y ∈ Rp | ∃{xk }k∈N ⊂ Rn such that K lim xk = ∞, k→∞ lim F (xk ) = y, k→∞ and lim νF (xk ) = 0} k→∞ ˜ ∞ (F ) is closed, and K ˜ ∞ (F ) = ∅ if F is a proper map in the sense that Clearly the set K lim x →∞ F (x) = ∞ 2.4 Newton polyhedra Throughout the text, we consider a fixed coordinate system x1 , , xn ∈ Rn Let J ⊂ {1, , n}, then we define RJ := {x ∈ Rn | xj = 0, for all j ∈ J} We denote by R+ the set of non-negative real numbers We also set Z+ := R+ ∩ Z If κ = (κ1 , , κn ) ∈ Zn+ , we denote by xκ the monomial xκ1 · · · xκnn and by |κ| the sum κ1 + · · · + κn Definition 2.4 A subset Γ ⊂ Rn+ is said to be a Newton polyhedron at infinity, if there exists some finite subset A ⊂ Zn+ such that Γ is equal to the convex hull in Rn of A ∪ {0} Hence we say that Γ is the Newton polyhedron at infinity determined by A and we write Γ = Γ(A) We say that a Newton polyhedron at infinity Γ ⊂ Rn+ is convenient if it intersects each coordinate axis in a point different from the origin, that is, if for any i ∈ {1, , n} there exists some integer mj > such that mj ej ∈ Γ, where {e1 , , en } denotes the canonical basis in Rn Given a Newton polyhedron at infinity Γ ⊂ Rn+ and a vector q ∈ Rn , we define d(q, Γ) := min{ q, κ | κ ∈ Γ}, ∆(q, Γ) := min{κ ∈ Γ | q, κ = d(q, Γ)} We say that a subset ∆ of Γ is a face of Γ if there exists a vector q ∈ Rn such that ∆ = ∆(q, Γ) The dimension of a face ∆ is defined as the minimum of the dimensions of the affine subspaces containing ∆ The faces of Γ of dimension are called the vertices of Γ We denote by Γ∞ the union of the faces of Γ which not contain the origin in Rn Let Γ1 , , Γp be a collection of p Newton polyhedra at infinity in Rn+ , for some p ≥ The Minkowski sum of Γ1 , , Γp is defined as the set Γ1 + · · · + Γp = {κ1 + · · · + κp | κi ∈ Γi , for all i = 1, , p} By definition, Γ1 + · · · + Γp is again a Newton polyhedron at infinity Moreover, by applying the definitions given above, it is easy to check that d(q, Γ1 + · · · + Γp ) = d(q, Γ1 ) + · · · + d(q, Γp ), ∆(q, Γ1 + · · · + Γp ) = ∆(q, Γ1 ) + · · · + ∆(q, Γp ), for all q ∈ Rn As an application of these relations, we obtain Lemma 2.3 Let ∆ be a face of Γ1 + · · · + Γp Then there exists a unique collection of faces ∆1 , , ∆p of Γ1 , , Γp , respectively, such that ∆ = ∆ + · · · + ∆p In particular, Γ∞ ⊂ Γ1,∞ + · · · + Γp,∞ Let f : Rn → R be a polynomial function Suppose that f is written as f = κ aκ xκ Then the support of f, denoted by supp(f ), is defined as the set of those κ ∈ Zn+ such that aκ = We denote the set Γ(supp(f )) by Γ(f ) This set will be called the Newton polyhedron at infinity of f The polynomial f is said to be convenient when Γ(f ) is convenient If f ≡ 0, then we set Γ(f ) = ∅ Note that, if f is convenient, then for each nonempty subset J of {1, , n}, we have Γ(f ) ∩ RJ = Γ(f |RJ ) The Newton boundary at infinity of f , denoted by Γ∞ (f ), is defined as the union of the faces of Γ(f ) which not contain the origin in Rn Let us fix a face ∆ of Γ∞ (f ) We define the principal part of f at infinity with respect to ∆, denoted by f∆ , as the sum of those terms aκ xκ such that κ ∈ ∆ Remark 2.4 By definition, for each face ∆ of Γ∞ there exists a vector q = (q1 , , qn ) ∈ Rn with minj=1, ,n qj < such that ∆ = ∆(q, Γ) 2.5 Non-degeneracy at infinity In [29] (see also [32]), Khovanskii introduced a condition of non-degeneracy of complex analytic maps F : (Cn , 0) → (Cp , 0) in terms of the Newton polyhedra of the component functions of F This notion has been applied extensively to the study of several questions concerning isolated complete intersection singularities (see for instance [9, 15, 22, 53]) We will apply this condition for real polynomial maps First we need to introduce some notation Let F := (f1 , , fp ) : Rn → Rp , ≤ p ≤ n, be a polynomial map Let Γ(F ) denote the Minkowski sum Γ(f1 ) + · · · + Γ(fp ), and we denote by Γ∞ (F ) the union of the faces of Γ(F ) which not contain the origin in Rn Let ∆ be a face of the Γ(F ) According to Lemma 2.3, let us consider the decomposition ∆ = ∆1 + · · · + ∆p , where ∆i is a face of Γ(fi ), for all i = 1, , p We denote by F∆ the polynomial map (f1,∆1 , , fp,∆p ) : Rn → Rp , and the Jacobian matrix of F∆ at x is denoted by DF∆ (x) Definition 2.5 We say that F is Khovanskii non-degenerate at infinity if and only if for any face ∆ of Γ∞ (F ), we have F∆−1 (0) ∩ {x ∈ Rn | rank(DF∆ (x)) < p} ⊂ {x ∈ Rn | x1 · · · xn = 0} The following result will be useful for our later analysis Theorem 2.3 Let F = (f1 , , fp ) : Rn → Rp be a polynomial map such that fi is convenient, for all i = 1, , p Suppose that F is Khovanskii non-degenerate at infinity Then ˜ ∞ (F ) = ∅ K ˜ ∞ (F ) = ∅; i.e., there exist a point y ∈ Rp and a Proof By contradiction, suppose that K sequence {xk }k∈N ⊂ Rn such that lim xk = ∞, k→∞ lim F (xk ) = y, k→∞ and lim νF (xk ) = k→∞ By definition, there exists a sequence λk := (λk1 , , λkp ) ∈ Rp , with we have for all k ≥ 1, p i=1 |λki | = 1, such that p λki ∇fi (xk ) k νF (x ) = i=1 By the Curve Selection Lemma at infinity (Lemma 2.1), there exist analytic curves ϕ(t) := (ϕ1 (t), , ϕn (t)) and λ(t) := (λ1 (t), , λp (t)), < t 1, such that (a1) limt→0 ϕ(t) = ∞; (a2) limt→0 F (ϕ(t)) = y ∈ Rp ; (a3) pi=1 |λi (t)| = 1; and p (a4) limt→0 i=1 λi (t)∇fi (ϕ(t)) = Let J := {j | ϕj ≡ 0} By Condition (a1), J = ∅ Thanks to Growth Dichotomy Lemma (Lemma 2.2), for each j ∈ J, we can expand the coordinate ϕj in terms of the parameter: say ϕj (t) = x0j tqj + higher order terms in t, where x0j = From Condition (a1), we get minj∈J qj < Recall that RJ := {κ := (κ1 , κ2 , , κn ) ∈ Rn | κj = for j ∈ J} Since fi is convenient, Γ(fi ) ∩ RJ = ∅ Let di be the minimal value of the linear function j∈J qj κj on Γ(fi ) ∩ RJ , and let ∆i be the (unique) maximal face of Γ(fi ) ∩ RJ where the linear function takes this value Since fi is convenient, di < and ∆i is a face of Γ∞ (fi ) Note that fi,∆i does not dependent on xj for all j ∈ J By a direct calculation, then fi (ϕ(t)) = fi,∆i (x0 )tdi + higher order terms in t, where x0 := (x01 , , x0n ) with x0j = for j ∈ J By Condition (a2) and di < 0, we have (3) fi,∆i (x0 ) = 0, for all i = 1, , p Let I := {i | λi ≡ 0} It follows from Condition (a3) that I = ∅ For i ∈ I, expand the coordinate λi in terms of the parameter: say λi (t) = λ0i tθi + higher order terms in t, where λ0i = For i ∈ I and j ∈ J we have ∂fi ∂fi,∆i di −qj (ϕ(t)) = (x )t + higher order terms in t ∂xj ∂xj It implies that λi (t) i∈I ∂fi (ϕ(t)) = ∂xj λ0i ∂fi,∆i di +θi −qj (x )t + higher order terms in t ∂xj λ0i ∂fi,∆i (x ) t −qj + higher order terms in t, ∂xj i∈I = i∈I where := mini∈I (di + θi ) and I := {i ∈ I | di + θi = } = ∅ Then by Condition (a4), we have for all j ∈ J, i∈I p ∂fi λi (t) (ϕ(t)) = ∂xj λi (t) i=1 ∂fi (ϕ(t)) → 0, ∂xj as t → There are two cases to be considered Case 1: ≤ qj∗ := minj∈J qj We have for all j ∈ J, λ0i i∈I ∂fi,∆i (x ) = 0, ∂xj which implies easily that   rank   x01 x01 ∂f1,∆1 (x0 ) ∂x1 ∂fp,∆p (x0 ) ∂x1 ··· x0n ··· ··· x0n ∂f1,∆1 (x0 ) ∂xn ∂fp,∆p (x0 ) ∂xn    < p  This, together with (3), contradicts with the assumption that the polynomial map F = (f1 , , fp ) is Khovanskii non-degenerate at infinity Case 2: > qj∗ := minj∈J qj It follows from Condition (a3) that θi ≥ for all i ∈ I and θi = for some i ∈ I Without lost of generality, we may assume that ∈ I and θ1 = Since f1 is convenient, for any j = 1, , n, there exists a natural number mj ≥ such that mj ej ∈ Γ∞ (f1 ) Then it is clear that qj mj ≥ d1 , for all j ∈ J On the other hand, we have d1 = d1 + θ1 ≥ min(di + θi ) = i∈I Therefore qj∗ mj∗ ≥ d1 ≥ > qj∗ Since qj∗ = minj∈J qj < 0, it implies that mj∗ < 1, which is a contradiction 10 Theorem 4.3 Let F = (f1 , , fp ) : Rn → Rp , ≤ p ≤ n, be a polynomial map such that fi is convenient, for all i = 1, , p Suppose that F is non-degenerate at infinity Let f (x) := maxi=1, ,p fi (x) and S := {x ∈ Rn | f (x) ≤ 0} = ∅ Then there exists a constant c > such that cd(x, S) ≤ [f (x)]+H(d,n,p) + [f (x)]+ for all x ∈ Rn , where d := maxi=1, ,p deg fi The following lemma is crucially used in the proof of Theorem 4.3 Lemma 4.2 Under the assumptions of Theorem 4.3, there exist some constants c > and R > such that mf (x) ≥ c for all x ≥ R Proof Suppose that by contradiction there exists a sequence {xk }k∈N ⊂ Rn such that lim xk = ∞, k→∞ and lim mf (xk ) = k→∞ We remark that the function mf (x) is semi-algebraic By Lemma 4.1 and by the Curve Selection Lemma at infinity, there exist a nonempty subset I˜ ⊂ {1, , p}, an analytic curve ˜ for < t ϕ(t) := (ϕ1 (t), , ϕn (t)) and some analytic functions λi (t), i ∈ I, 1, such that (d1) (d2) (d3) (d4) limt→0 ϕ(t) = ∞; ˜ and fi (ϕ(t)) < f (ϕ(t)) for i ∈ I; ˜ fi (ϕ(t)) = f (ϕ(t)) for i ∈ I, ˜ and λi (t) ≥ for all i ∈ I, i∈I˜ λi (t) = limt→0 mf (ϕ(t)) = limt→0 i∈I˜ λi (t)∇fi (ϕ(t)) = Let J := {j | ϕj ≡ 0} By Condition (d1), J = ∅ In view of Growth Dichotomy Lemma, for j ∈ J, we can expand the coordinate ϕj in terms of the parameter: say ϕj (t) = x0j tqj + higher order terms in t, where x0j = From Condition (d1), we get qj∗ := minj∈J qj < Note that ϕ(t) ctqj∗ + o(tqj∗ ) as t → 0, for some c > = Since fi is convenient, Γ(fi ) ∩ RJ = ∅ Let di be the minimal value of the linear function J J j∈J qj κj on Γ(fi ) ∩ R , and let ∆i be the (unique) maximal face of Γ(fi ) ∩ R where the linear function takes this value Since fi is convenient, di < and ∆i is a face of Γ∞ (fi ) Note that fi,∆i does not dependent on xj for all j ∈ J By a direct calculation, then fi (ϕ(t)) = fi,∆i (x0 )tdi + higher order terms in t, where x0 := (x01 , , x0n ) with x0j = for j ∈ J Let I := {i ∈ I˜ | λi ≡ 0} It follows from Condition (d3) that I = ∅ For i ∈ I, expand the coordinate λi in terms of the parameter: say λi (t) = λ0i tθi + higher order terms in t, 20 where λ0i = For i ∈ I and j ∈ J we have ∂fi ∂fi,∆i di −qj (ϕ(t)) = (x )t + higher order terms in t ∂xj ∂xj It implies that λi (t) i∈I ∂fi (ϕ(t)) = ∂xj λ0i ∂fi,∆i di +θi −qj (x )t + higher order terms in t ∂xj λ0i ∂fi,∆i (x ) t −qj + higher order terms in t, ∂xj i∈I = i∈I where := mini∈I (di + θi ) and I := {i ∈ I | di + θi = } = ∅ There are two cases to be considered Case 1: ≤ qj∗ := minj∈J qj We deduce from Condition (d4) that λ0i i∈I ∂fi,∆i (x ) = 0, ∂xj j ∈ J for all Hence rank x0j ∂fi,∆i (x0 ) ∂xj < #I i∈I ,1≤j≤n On the other hand, it follows from Lemma 2.4 that the map FI is Khovanskii non-degenerate at infinity Therefore there exists an index i0 ∈ I such that fi0 ,∆i0 (x0 ) = Then, by ˜ Condition (d2), we have for all i ∈ I, f (ϕ(t)) = fi (ϕ(t)) = fi0 (ϕ(t)) = fi0 ,∆i0 (x0 )tdi0 + higher order terms in t By taking the derivative in t of the function (f ◦ ϕ)(t), we deduce that d(f ◦ ϕ)(t) d(fi ◦ ϕ)(t) = = dt dt ∇fi (ϕ(t)), dϕ(t) dt , for all ˜ i ∈ I By Condition (d3), then d(f ◦ ϕ)(t) = dt λi (t) i∈I˜ d(f ◦ ϕ)(t) = dt λi (t)∇fi (ϕ(t)), i∈I˜ Thus d(f ◦ ϕ)(t) dt ≤ mf (ϕ(t)) dϕ(t) , dt which implies that mf (ϕ(t)) ≥ c tdi0 −qj∗ + higher order terms in t, 21 dϕ(t) dt for some c > But this inequality contradicts Condition (d4) since di0 ≤ di0 + θi0 = ≤ qj∗ Case 2: > qj∗ := minj∈J qj With a similar argument as in the proof of Theorem 2.3, it is not hard to get a contradiction By Lemma 4.2, the semi-algebraic function f (x) := maxi=1, ,p fi (x) satisfies the PalaisSmale condition at each t ∈ R Then, it follows from Theorem 4.2 that a global error bound result holds for f Before proving Theorem 4.3 which establishes that a H¨older-type global error bound holds, we give an error bound result on a bounded region Lemma 4.3 Let S denote the set of x in Rn satisfying f1 (x) ≤ 0, , fp (x) ≤ 0, where each fi is a real polynomial Let R be a positive number such that S contains an element x with x ≤ R Then, there exists a constant c > such that cd(x, S) ≤ [f (x)]+H(d,n,p) for all x with x ≤ R Here f (x) := maxi=1, ,p fi (x) and d := maxi=1, ,p deg fi Proof Consider the polynomial h : Rn × Rp given by h(x, z) := (f1 (x) + z12 )2 + · · · + (fp (x) + zp2 )2 , where z := (z1 , , zp ) Let S¯ := {(x, z) ∈ Rn+p | h(x, z) = 0} It is easy seen that x∈S ⇔ ¯ (x, z) ∈ S, with zi := [−fi (x)]+ , i = 1, , p Further, since S = ∅ is nonempty, it follows that S¯ = ∅ Let R := max{ z | zi := [−fi (x)]+ , i = 1, , p, x ≤ R} > By [56, Theorem 2.2] (see also [34]), it is not hard to check that ¯ ≤ h(x, z) 2H(d,n,p) c1 d((x, z), S) for all x ≤ R and z ≤ R , for some c1 > Now by a similar argument as in the proof of [41, Theorem 2.2], one can get the conclusion easily In fact, given any x ∈ Rn with x ≤ R, let z ∈ Rp be given by zi := [−fi (x)]+ , i = 1, , p Then, by definition, we can see that z ≤ R and fi (x) + zi2 = fi (x) + [−fi (x)]+ = [fi (x)]+ ≤ [f (x)]+ , 22 for i = 1, , p It implies that p (fi (x) + zi2 )2 ≤ p[f (x)]2+ h(x, z) = i=1 Therefore c1 x − x ∗ ¯ ≤ h(x, z) 2H(d,n,p) ≤ c1 d((x, z), S) ≤ (p[f (x)]2+ ) 2H(d,n,p) , from which the desired result follows immediately Now, we are in position to finish the proof of Theomrem 4.3 Proof of Theorem 4.3 Let us again consider the continuous semi-algebraic function f+ : Rn → R, x → max{f (x), 0} By definition, if f (x) > then f+ (x) = f (x), ∂f+ (x) = ∂f (x) and mf+ (x) = mf (x) In view of Lemma 4.2, there exist c1 > 0, δ > and R > such that mf+ (x) ≥ c1 (4) x ∈ f+−1 ((0, +∞)) and for all x ≥ R Thanks to Lemma 4.3, there is a constant c2 > such that c2 d(x, S) ≤ f+ (x) H(d,n,p) (5) for all x ≤ R Let x ∈ Rn be such that x ∈ f+ −1 ((0, ∞)) and x > R By [11, Corollary 4.1], there exists a maximal absolutely continuous curve u : [0, ∞) → Rn of the dynamical system ∈ u(s) ˙ + ∂[f+ (u(s))] satisfying u(0) = x In addition, the function s → f+ ◦ u(s) is absolutely continuous and strictly decreasing on [0, ∞) By [11, Corollary 4.2], we have for almost all s ∈ [0, ∞), u(s) ˙ = mf+ (u(s)) and d (f+ ◦ u)(s) = −[mf+ (u(s))]2 ds We have the following remark Suppose that f+ (u(s)) > and u(s) ≥ R for all s ∈ [t1 , t2 ], for some ≤ t1 < t2 It follows from the inequality (4) that t2 f+ (u(t1 )) − f+ (u(t2 )) = − t1 t2 ≥ d (f+ ◦ u)(s)ds = ds t2 [mf+ (u(s))]2 ds t1 t2 c1 mf+ (u(s))ds = t1 c1 u(s) ˙ ds, t1 which yields (6) f+ (u(t1 )) − f+ (u(t2 )) ≥ c1 | u(t1 ) − u(t2 ) | 23 Hence the curve u has finite length, and so it is bounded In view of [11, Theomrem 4.5], there exists the limit a := lims→∞ f (u(s)) In addition, we have mf (a) = Let t := inf{s | u(s) > R} There are two cases to be considered Case 1: t = ∞; i.e., u(s) > R for all s ≥ Since mf (a) = 0, it follows from the inequality (4) that f+ (a) = Therefore, by (6), we obtain f+ (x) = f+ (x) − f+ (a) ≥ c1 u(0) − a = c1 x − a ≥ c1 d(x, S) Case 2: t < ∞ We have u(t) = R Then it follows from (4), (5) and (6) that d(x, S) ≤ d(x, u(t)) + d(u(t), S) f+ (x) − f+ (u(t)) (f+ (u(t))) H(d,n,p) + ≤ c1 c2 f+ (x) (f+ (x)) H(d,n,p) ≤ + c1 c2 In summary, in both cases, we have cd(x, S) ≤ f+ (x) + (f+ (x)) H(d,n,p) , where c := min{c1 , c2 } This, together with (5), completes the proof of Theorem 4.3 Openness and density Let Γ1 , , Γp be a collection of p Newton polyhedra in Rn+ for some ≤ p ≤ n Let Γ denote the Minkowski sum Γ1 + · · · + Γp Then we define D(Γ) := {c := (c1 , , cp ) ∈ Rm1 × · · · × Rmp | ci = (ci,κ ), ci,κ xκ ∈ R[x], Γ(fi ) = Γi , and fi,ci (x) := fi (x, ci ) = κ Fc (x) := (f1,c1 (x), , fp,cp (x)) is non-degenerate at infinity} For each a subset I = {i1 , , iq } ⊂ {1, , p}, we denote DI (Γ) := {c := (c1 , , cp ) ∈ Rm1 × · · · × Rmp | ci = (ci,κ ), ci,κ xκ ∈ R[x], Γ(fi ) = Γi , and fi,ci (x) := fi (x, ci ) = κ FI,c (x) := (fi1 ,ci1 (x), , fiq ,ciq (x)) is Khovanskii non-degenerate at infinity} It follows from Lemma 2.4 that D(Γ) = ∩I⊂{1, ,p} DI (Γ) 24 The main result of this section is the following Theorem 5.1 The set D(Γ) ⊂ Rm1 × · · · × Rmp is an open and dense semi-algebraic set Proof Since the number of subsets of {1, , p} is finite and a finite intersection of open and dense semi-algebraic sets is open and dense semi-algebraic, it is sufficient to prove that for every I ⊂ {1, , p}, the set DI (Γ) is open dense semi-algebraic For simplifying the notation, we will only consider the case I = {1, , p}, the other cases are completely similar The proof follows immediately from Claims 5.1 and 5.2 below Claim 5.1 The set D{1, ,p} (Γ) is open and semi-algebraic Proof The idea of the proof took from [52, Appendix] For every face ∆ = ∆1 + · · · + ∆p ⊂ Γ∞ , we define: V (∆) := {(x, c1 , , cp ) ∈ Rn × Rm1 × · · · × Rmp | f1,∆1 (x, c1 ) = · · · = fp,∆p (x, cp ) = 0,  ∂f  1,∆1 ∂f1,∆1 x01 ∂x (x, c ) (x, c ) · · · x 1 n ∂xn     = p, rank  ···  ∂fp,∆p ∂fp,∆p x1 ∂x1 (x, cp ) · · · xn ∂xn (x, cp ) V (∆)∗ := V (∆) ∩ {(x, c1 , , cp ) ∈ Rn × Rm1 × · · · × Rmp | x1 · · · xn = 0} Note that V (∆) is closed and that V (∆)∗ = V (∆) Let us consider the union V ∗ := V (∆)∗ and the projection π : Rn × Rm1 × · · · × Rmp → Rm1 × · · · × Rmp Showing that ∆⊂Γ∞ D{1, ,p} (Γ) is an open set means to prove that its complement W := π(V ∗ ) is a closed set One observes that W is a semi-algebraic set, since it is the projection of a semi-algebraic set (Theorem 2.1) Let (c01 , , c0p ) ∈ W By the Curve Selection Lemma at infinity, there exists a face ∆ of Γ∞ and a real analytic path (ϕ(t), c1 (t), , cp (t))) ∈ V (∆)∗ defined on a small enough interval (0, ) such that (f1) limt→0 ci (t) = c0i , i = 1, , p; and (f2) either limt→0 ϕ(t) = ∞ or limt→0 ϕ(t) = a ∈ V (∆) Let us expand the coordinate ϕj , j = 1, , n, in terms of the parameter, say ϕj (t) = x0j tqj + higher order terms in t, where x0j = and qj ∈ Z According to Lemma 2.3, let us consider the decomposition ∆ = ∆1 + · · · + ∆p , where ∆i is a face of Γ(fi ), for all i = 1, , p Let di be the minimal value of the linear function n qj κj on ∆i , and let ∆i be the (unique) maximal face of ∆i where the linear function j=1 25 takes this value Clearly, ∆i is a face of Γ∞ (fi ) By a direct calculation, then fi,∆i (ϕ(t), ci (t)) = fi,∆i (x0 , ci (t))tdi + higher order terms in t, ϕj (t) ∂fi,∆i ∂fi,∆i (ϕ(t), ci (t)) = x0j (x , ci (t))tdi + higher order terms in t, ∂xj ∂xj where x0 := (x01 , , x0n ) ∈ (R \ 0)n By taking the limit ci (t) → c0i , i = 1, , p, and focusing on the first terms of the above expansions we get that (x0 , c01 , , c0p ) ∈ V (∆) ⊂ V ∗ , where ∆ := ∆1 + · · · + ∆p Thus (c01 , , c0p ) ∈ W, which concludes the proof that W = W Claim 5.2 The set D{1, ,p} (Γ) contains an open and dense semi-algebraic set Proof For a polynomial f , we denote x∇f (x) := (x1 ∂f ∂f (x), , xn (x)) and for a poly∂x1 ∂xn nomial map F = (f1 , , fp ) we let xDF (x) := (x1 D1 F (x), , xn Dn F (x)) = xj ∂fi (x) ∂xj i=1, ,p,j=1, ,n First of all, since the number of faces of Γ is finite and a finite intersection of open and dense semi-algebraic sets is open and dense semi-algebraic, it suffices to consider the case where Γ is a face So, let us fix a face ∆ of Γ∞ By Lemma 2.3, the face ∆ admits a unique decomposition ∆ = ∆1 + · · · + ∆p , where ∆i is a compact face of Γi for all i = 1, , p For simplifying the notations, suppose that Γi = ∆i for i = 1, , p Secondly, it is clear that if there exists an index i0 such that mi0 = 1; i.e., fi, ci0 is a monomial, then {fi, ci0 = 0} ⊂ {x1 xn = 0} for ci0 = So D{1, ,p} (Γ) ⊃ (R∗ )m1 × · · · × (R∗ )mi0 × · · · × (R∗ )mp and the problem is trivial So we may assume that mi > for every i = 1, , p Let d := dim ∆ There are two cases to be considered Case 1: d < p By [21, Exercises of page 48], there exist n vectors q , , q n ∈ Zn , with det(q , , q n ) = 1, such that ∆ is contained in the plane L := {κ ∈ Rn | q j , κ = dj , j = 1, , n − d} We need the following lemma Lemma 5.1 For each i there exist dij , j = 1, , n − p, such that ∆i ⊂ Li , where Li := {κ ∈ Rn | q j , κ = dij , j = 1, , n − d}, i.e., ∆i is contained in a plane which is parallel to L 26 Proof For k = 1, , p, let κk ∈ ∆k We fix an index i and let κ ∈ ∆i be such that κ = κi Set dij := q j , κi By definition, γ := pk=1 κk ∈ ∆ and γ := k=i κk + κ ∈ ∆ Hence q j , γ = q j , γ = dj , and so q j , γ−γ = Thus q j , κ−κi = 0; i.e., q j , κ = q j , κi = dij Consequently, κ ∈ Li The lemma is proved For each j = 1, , n, let us write q j := (qj1 , , qjn ) Consider the following change of coordinates  q(n−d)1 q11 qn1    x1 = u1 un−d un , (7)   q (n−d)n  x = uq1n u uqnn n n n n−d It follows from Lemma 5.1 that for each κ ∈ ∆i ∩ Zn , we have Aκ = (di1 , , din−d , γκ ), for some γκ ∈ Zd So in the system of coordinates u1 , , un , the polynomial fi has the form d i(n−d) fi (u) = ud1i1 un−d (8) ci,κ u γκ , κ∈∆i ∩Zn where u = (un−d+1 , , un ) Set (9) ci,κ u γκ gi (u ) = gi,ci (u ) = κ∈∆i ∩Zn Since A := (qij )i,j=1, ,n is an integer matrix and det(A) = 1, the monomial map (7) has the monomial inverse map, given by A−1 Hence the system f1,c1 = · · · = fp,cp = has solutions in (R∗ )n if and only if the system g1,c1 = · · · = gp,cp = has solutions in (R∗ )d Consider the map G: Rp Rm1 × · · · × Rmp × (R∗ )d → (c1 , , cp , u ) → (g1,c1 (u ), , gp,cp (u )) For i = 1, , p, let κi ∈ ∆i ∩ Zn Then the Jacobian matrix DG of G contains the following diagonal matrix    ∂G =   ∂(c1,κ1 , , cp,κp ) u γκ1 u γκp    ∂G is of ∂(c1,κ1 , , ∂cp,κp ) {0} (in Rp ) By the Transversality Theorem Since u ∈ (R∗ )d , it follows that u γκi = for every i So the matrix rank p Hence DG is of rank p, which yields G (Theorem 2.2), the set P1 := {c = (c1 , , cp ) ∈ Rm := Rm1 × · · · × Rmp | G(c, ) {0}} is an open and dense semi-algebraic set in Rm Since d < p, the map G(c, ) : (R∗ )d → Rp is transversal to {0} if and only if ImG(c, ) ∩ {0} = ∅ We deduce, for each c ∈ P1 , that 27 {Gc := G(c, ) = 0} ∩ (R∗ )d = ∅, and hence {Fc := (f1,c1 , , fp,cp ) = 0} ∩ (R∗ )n = ∅ This implies that Fc is Khovanskii non-degenerate at infinity Case 2: d ≥ p We first remark that we may assume that d = n In fact, suppose that d < n Under changing of coordinates (7), the polynomials fi,ci and gi,ci have the forms (8) and (9), respectively Let Fc := (f1,c1 , , fp,cp ) and Gc := (g1,c1 , , gp,cp ) We have {x ∈ (R∗ )n | Fc (x) = and rank[xDFc (x)] < p} = ∅ {u ∈ (R∗ )d | Gc (u ) = and rank[xDGc (u )] < p} = ∅ only if But Gc is a polynomial map in d variables Therefore, the problem is reduced to the case d = n For each i = 1, , p, let v i1 , , v iri be the vertices of ∆i Note that ri > for every i ij Let cij be the coefficient of the monomial xv in fi Let wik := v ik − v iri for k = 1, , ri − 1, and consider the system of vectors W := {w11 , , w1(r1 −1) , , wp1 , , wp(rp −1) } ⊂ Rn First of all, we prove the following lemma Lemma 5.2 With the above notations, we have rank(W) = d = dim ∆ Proof Let a, b be two arbitrary points of ∆ Since ∆ = ∆1 + · · · + ∆p , there exist , bi ∈ ∆i , i = 1, , p, such that a = a1 + · · · + ap , b = b1 + · · · + bp Note that ri = ri λij v ij , with λij ≥ and λij = 1, j=1 j=1 ri b i ri ij = µij v , with µij ≥ and µij = j=1 j=1 Hence ri −1 ri b i − = (µij − λij )v ij = j=1 (µij − λij )v ij + (µiri − λiri )v iri j=1 ri −1 ri −1 ij (µij − λij )v iri (µij − λij )v − = j=1 j=1 ri −1 (µij − λij )wij = j=1 28 Consequently, b − a = pi=1 (bi − ) is a linear combination of vectors of W Since a, b can be chosen arbitrarily, the lemma follows Now we will prove Claim 5.2 (Case 2) by induction on p, the number of polynomials Firstly, consider the case p = Note that, by assumption, dim ∆1 = n Consider the map Φ : Rm1 × (R∗ )n → Rn+1 , (c1 , x) → (x∇f1,c1 (x), f1,c1 (x)) The Jacobian matrix DΦ of Φ contains the following matrix ∂Φ = ∂(c1,1 , , c1,r1 ) 1(r −1) 11 1r xv v 11 xv v 1(r1 −1) xv v 1r1 1r 1(r −1) 11 xv xv xv where v 1j are written as column vectors The rank of , ∂Φ is equal to the rank ∂(c1,1 , , c1,r1 ) of the following matrix M1 = v 11 v 1(r1 −1) v 1r1 1 By some linear operations on the columns of M1 , we obtain the following matrix with the same rank M2 = v 11 − v 1r1 v 1(r1 −1) − v 1r1 v 1r1 = w11 w1(r1 −1) v 1r1 But we know from Lemma 5.2 that rank{w11 , , w1(r1 −1) } = dim ∆1 = d = n Hence rankM2 = n + So rank(DΦ) = n + Consequently, we get Φ the Transversality Theorem, the set P2 := {c1 ∈ Rm1 | Φ(c1 , ) {0} (in Rn+1 ) By {0}} is an open and dense semi-algebraic in Rm1 Note that the map Φ(c1 , ) : (R∗ )n → Rn+1 is transversal to {0} if and only if ImΦ(c1 , ) ∩ {0} = ∅ Hence, for c1 ∈ P2 , we have {Φ(c1 , ) = 0} = ∅ This completes the proof in the case p = Now by induction, assume that for each l = 1, , p, the set Ul contains an open and dense semi-algebraic set in Rm , where Ul := D{1, ,p}\{l} (Γ) = {c := (c1 , , cp ) ∈ Rm | ci = (ci,κ ), ci,κ xκ , Γ(fi ) = Γi , and fi,ci (x) := fi (x, ci ) = κ Fcl (x) := (f1,c1 (x), , fl−1,cl−1 (x), fl+1,cl+1 (x), , fp,cp (x)) is Khovanskii non-degenerate at infinity} 29 Consider the map Ψ: U1 ∩ · · · ∩ Up × (R∗ )n × (Rp − {0}) → Rn × Rp , f1,c1 (x), , fp,cp (x)) p (c1 , , cp , x , → ( λ) λi x∇fi,ci (x) i=1 Note that if (c, x, λ) ∈ Ψ−1 (0), then λ1 λp = In fact, if λl = for some l, then l i=l λi x∇fi,ci (x) = Hence, the map Fc is not Khovanskii non-degenerate at infinity, which contradicts the fact that (c1 , , cp ) ∈ U1 ∩ · · · ∩ Up ⊂ Ul ∂Ψ The Jacobian matrix DΨ of Ψ contains the matrix M3 := = ∂[(c1,1 , , c1,r1 ), , (cp,1 , , cp,rp )]  p(rp −1) prp p1 11 1(r −1) 1r v p(rp −1) λp xv v prp λ1 xv v 11 λ1 xv v 1(r1 −1) λ1 xv v 1r1 λp xv vp1 λp xv 1r 1(r −1) 11  λ xv λ1 xv 0  λ xv   0 0    λ p xv p1 λp xv p(rp −1) λp xv where v ij are written as column vectors ij If (c, x, λ) ∈ Ψ−1 (0), we know that λi xv = for i = 1, , p, and j = 1, , ri Hence, by some multiplications on the columns of M3 , we obtain the following matrix which has the same rank   11 v v 1(r1 −1) v 1r1 v p1 v p(rp −1) v prp   1 0      0 0 M4 =        0 1 By some linear operations on the columns of M4 , we obtain  11  v − v 1r1 v 1(r1 −1) − v 1r1 v 1r1 v p1 − v prp v p(rp −1) − v prp v prp   0      0 0 M5 =        0  11  w w1(r1 −1) v 1r1 wp1 wp(rp −1) v prp   0      0 0 =        0 30 prp     ,    By rearranging the columns of M5 , we get  w11 w1(r1 −1) wp1 wp(rp −1) v 1r1 v prp   M6 =          But we know from Lemma 5.2 that rank{w11 , , w1(r1 −1) , , wp1 , , wp(rp −1) } = dim ∆ = d = n So rankM6 = n + p on Ψ−1 (0) Thus DΨ is of maximal rank on Ψ−1 (0) Hence Ψ {0} (in ∈ Rn+p ) Note that U1 ∩ · · · ∩ Up contains an open and dense semi-algebraic set in Rm Again, by the Transversality Theorem, the set P3 := {c ∈ U1 ∩ · · · ∩ Up | Ψ(c, ) {0}} is open and dense semi-algebraic Since Ψ(c, ) : (R∗ )n × (Rp − {0}) → Rn × Rp is a map between two manifolds of same dimension, the transversality condition implies that Ψ(c, ) is a local diffeomorphism on (Ψ(c, ))−1 (0) for each c ∈ P3 Let c ∈ P3 If (Ψ(c, ))−1 (0) = ∅, there exists (x, λ) ∈ (R∗ )n+p such that Ψ(c, x, λ) = Moreover, for every t ∈ R \ {0}, Ψ(c, x, tλ) = So Ψ(c, ) is not a local diffeomorphism at (x, λ), which is a contradiction Hence (Ψ(c, ))−1 (0) = ∅ Consequently, the map Fc is Khovanskii 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systems, Optimization, 31 (1994), 1-12 [61] W H Yang, Error bounds for convex polynomials, SIAM J Optim., 19 (2009), 1633-1647 Institute of Mathematics, 18, Hoang Quoc Viet Road, Cau Giay District 10307, Hanoi, Vietnam E-mail address: dstiep@math.ac.vn Institute of Mathematics, 18, Hoang Quoc Viet Road, Cau Giay District 10307, Hanoi, Vietnam E-mail address: hhvui@math.ac.vn Department of Mathematics, University of Dalat, Phu Dong Thien Vuong, Dalat, Vietnam E-mail address: sonpt@dlu.edu.vn 34 [...]... Pha.m, An explicit bound for the Lojasiewicz exponent of real polynomials, Kodai Mathematical Journal, (2012) [57] P J Rabier, Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds, Annals of Math 146 (1997), 647-691 33 [58] S M Robinson, An application of error bounds for convex programming in a linear space, SIAM J Control 13, 271-273 (1975) [59] R T Rockafellar, and. .. number for convex inequality systems and global error bounds for analytic systems, Math Program 83, 263-276 (1998) [17] S T Dinh, H V H` a, and N T Thao, Lojasiewicz inequality for polynomial functions on non compact domains, International Journal of Mathematics, Vol 23, No 4 (2012) 1250033 (28 pages), DOI: 10.1142/S0129167X12500334 [18] L van den Dries and C Miller, Geometric categories and o-minimal... Variational Analysis, Grundlehren Math Wiss., 317, Springer, New York, 1998 [60] T Wang and J S Pang, Global error bounds for convex quadratic inequality systems, Optimization, 31 (1994), 1-12 [61] W H Yang, Error bounds for convex polynomials, SIAM J Optim., 19 (2009), 1633-1647 Institute of Mathematics, 18, Hoang Quoc Viet Road, Cau Giay District 10307, Hanoi, Vietnam E-mail address: dstiep@math.ac.vn... Goresky, and R MacPherson, Stratified Morse theory, Springer, 1988 [24] V Guillemin, and A Pollack, Differential topology, Prentice-Hall, 1974 [25] H V H` a, and T S Pha.m, Solving polynomial optimization problems via the truncated tangency variety and sums of squares, J Pure Appl Algebra, 213 (2009), 2167-2176 [26] H V H` a, Global H¨ olderian error bound for non-degenerate polynomials, Submitted for publication,... 1968 [49] A N´emethi, and A Zaharia, Milnor fibration at infinity, Indag Math., 3 (1992), 323-335 [50] K F Ng, and X Y Zheng, Global error bounds with fractional exponents, Math Program., Ser B, 88 (2000), 357-370 [51] H V Ngai and M Thera, Error bounds for differentiable convex inequality systems in Banach spaces, Math Program., Ser B, 104 (2005), No.2-3, 465-482 [52] M Oka, On the bifurcation of the... the map Fc is Khovanskii non-degenerate at infinity Acknowledgments This research was performed while the authors had been visiting at Vietnam Institute for Advanced Study in Mathematics (VIASM) The authors would like to thank the Institute for hospitality and support References [1] D D’Acunto, and K Kurdyka, Explicit bounds for the Lojasiewicz exponent in the gradient inequality for polynomials, Ann... publication, 2012 [27] A J Hoffman, On approximate solutions of linear inequalities, Journal of Research of the National Bureau of Standards, 49 (1952), 263-265 [28] L H¨ ormander, On the division of distributions by polynomials, Ark Mat 3 N 53 (1958), 555-568 [29] A G Khovanskii, Newton polyhedra and toroidal varieties, Funct Anal Appl 11 (1978) 289-296 [30] D Klatte, Hoffman’s error bound for systems. .. On the asymptotic well behaved functions and global error bound for convex polynomials, SIAM J Optim., 20 (2010), No.4, 1923-1943 [37] G Li, Global error bounds for piecewise convex polynomials, Math Program., Ser .A, DOI 10.1007/s10107-011-0481-z [38] W Li, Error bounds for piecewise convex quadratic programs and applications, SIAM J on Control and Optimization, 33 (1995), 1511-1529 [39] S Lojasiewicz,... and S Simon, Semialgebraic Sard theorem for generalized critical values, J Differential Geom., 56, (2000) 67-92 32 [34] K Kurdyka, and S Spodzieja, Separation of real algebraic sets and the Lojasiewicz exponent, Preprint UL, 2011/7 [35] A S Lewis and J S Pang, Error bounds for convex inequality systems, Generalized Convexity, Generalized Monotonicity, J P Crouzeix, J E Martinez-Legaz and M.Volle (eds)... of the multiplicity and topology of the Newton boundary, J Math Soc Japan 31(3) (1979), 435-450 [53] M Oka, Non-degenerate complete intersection singularity, Actualit´es Math´ematiques, Hermann, Paris, 1997 [54] J S Pang, Error bounds in Mathematical Programming, Math Program., Ser.B, 79 (1997), 299-332 [55] A F Perold, generalization of the Frank- Wolfe Theorem, Mathematical Programming, vol 18, pp