Polynomial reformulation of the Kuo criteria for v-sufficiency of map-germs

In the paper a set of necessary and sufficient conditions for \textit{v-}sufficiency (equiv. \textit{sv-}sufficiency) of jets of map-germs $f:(\mathbb{R}^{n},0)\to (\mathbb{R}^{m},0)$ is proved which generalize both the Kuiper-Kuo and the Thom conditions in the function case ($m=1$) so as the Kuo conditions in the general map case ($m>1$). Contrary to the Kuo conditions the conditions proved in the paper do not require to verify any inequalities in a so-called horn-neighborhood of the (a'priori unknown) set $f^{-1}(0)$. Instead, the proposed conditions reduce the problem on \textit{v-}sufficiency of jets to evaluating the local {\L}ojasiewicz exponents for some constructively built polynomial functions.


1.
Introduction. In theory of dynamical systems and nonlinear analysis quite a number of problems depending on parameters require analyzing the structure of the set of solutions of nonlinear equations, the number of variables in which exceeds the number of equations. As a rule, arising equations are rather complicated for investigation and need to be simplified in one or another way. Clearly, such "simplification" may lead as to correct conclusions about the structure of solutions as to wrong ones. Often, small solutions of equations are of interest. In this case one of the most popular methods of simplification of equations is their truncation, when one casts out high order terms in power-series expansions of the corresponding equations. In the paper polynomial necessary and sufficient conditions are proved allowing to judge in which cases, in analysis of systems of real nonlinear equations of finite smoothness, truncation is permissible.
Given a map f : R n → R m with f (0) = 0, let us consider the set of solutions of the equation Even locally, this set is very complicated in general. As usual, the map f is called C k -smooth if all its components have continuous partial derivatives up to the order k inclusive. If f ∈ C k in a neighborhood of the origin then for each r ≤ k it is defined the r-th Taylor polynomial f (r) (x) of f (x) about the point x = 0 which will be called the r-truncation of f (x). Transition from equation (1) to the truncated equation is similar to the first-approximation method in the theory of stability and to the method of studying bifurcations by the passage to linearized equations in nonlinear analysis, and so on. As demonstrates the next example the sets of solutions of equations (1) and (2) may be topologically different.
Example. Let us discard in the next equations the terms of order higher than 4, that is perform the 4-truncation of the left-hand parts. Then the truncated equation has a single solution, x 1 = x 2 = 0. The first of the full equations also has the same single solution, x 1 = x 2 = 0, while the second of the full equations has a continuum of solutions, x 1 = x 2 2 . Thus, truncation of equations is not always permissible.
Therefore it is natural to ask when the structure of the zero-set of the truncated map f (r) is similar to that of the full map f . This problem concerns the property of sufficiency of jets. Roughly speaking, sufficiency of jets is the property that all maps with the same truncation have the same structure.
Following to [3] we recall briefly some definitions and results on sufficiency of jets. Let E [k] (n, m) denote the set of C k map-germs f : (R n , 0) → (R m , 0). Given r ≤ k, let j r f (0) denote the r-jet of f ∈ E [k] (n, m) at 0 ∈ R n which can be identified with the polynomial f (r) , and let J r (n, m) denote the set of r-jets in E [k] (n, m). We say f, g ∈ E [k] (n, m) are C 0 -equivalent, if there is a local homeomorphism h : We further say f, g ∈ E [k] (n, m) are vequivalent (resp. sv-equivalent), if f −1 (0) is homeomorphic to g −1 (0) as germs at 0 ∈ R n (resp. there is a local homeomorphism h : (R n , 0) → (R n , 0) such that h(f −1 (0)) = g −1 (0)). Given r ≤ k, we call an r-jet w ∈ J r (n, m) C 0 -sufficient (resp. v-sufficient, sv-sufficient) in E [k] (n, m), if any two maps f, g ∈ E [k] (n, m) with j r f (0) = j r g(0) = w are C 0 -equivalent (resp. v-equivalent, sv-equivalent).
Clearly, C 0 -sufficiency of jets implies sv-sufficiency, while the latter implies vsufficiency. In fact, according to D.J.A. Trotman and L.C. Wilson [25], v-sufficiency is equivalent to sv-sufficiency.
Concerning C 0 -sufficiency of jets in the function case (i.e. m = 1), we have Kuiper [17], T.-C. Kuo [18], J. Bochnak & S. Lojasiewicz [5]). 1) if and only if there are positive numbers C, ε such that For f ∈ E [r+1] (n, 1), the jet j r f (0) is C 0 -sufficient in E [r+1] (n, 1) if and only if there are numbers C, δ, ε > 0 such that K. Bekka and S. Koike [3] proved that the Kuiper-Kuo condition (3) is equivalent to the following Thom condition: there are numbers K, ε > 0 such that Verification of the Kuiper-Kuo conditions (3) and (4), so as of the Thom condition (5), may be reduced to the problem on evaluation of the rate of growth of a polynomial about one of its roots, which is equivalent to calculation of the socalled local Lojasiewicz exponents of a polynomial. Recall, that according to the Lojasiewicz theorem [21,22,23] for any polynomial p : R n → R with p(0) = 0 there are constants C, κ > 0 such that |p(x)| ≥ C|x| κ in a neighborhood of the zero root. The least κ for which the above inequality holds is called the local Lojasiewicz exponent for p and is denoted by L 0 (p). If the zero root of p is isolated then such a least value of κ exists and is rational [11,21,22,23]. Moreover, in this case L 0 (p) ≤ (d − 1) n + 1 [12] where d is the degree of p. There is quite a number of publications devoted to evaluation of the Lojasiewicz exponent, see, e.g., [1,6,7,9,10,12,13,20] and the bibliography therein.
In the above theorem, H s (f ; σ) denotes the horn-neighbourhood of f −1 (0), where V i is the span of the v j 's, j = i. Unfortunately, verification of the Kuo conditions (6) and (7) is not as "simple" as verification of the Kuiper-Kuo conditions (3), (4) or the Thom condition (5). The first problem here, not the major one, is that the function D(v 1 , . . . , v m ) is not defined explicitely, by a "simple" formula. This causes problems in practical evaluation of D(v 1 , . . . , v m ). The second problem, which is more serious, is that one need evaluate the values of D(grad f   m (x)) not in a neighborhood of the origin but in horn-neighbourhoods of the sets (f (r) ) −1 (0) or g −1 (0) which are a'priory unknown in general. At last, in the case of v-sufficiency in E [r+1] (n, m) one need to verify condition (7) not for a single horn-neighbourhood but for a variety of horn-neighbourhoods defined for infinite number of polynomial maps g of degree r + 1 satisfying j r g(0) = j r f (0).
In the above theorem (df ) * (x) denotes the matrix conjugate to df (x). Clearly, the matrix (df ) * (x) consists of m column vectors grad f j (x), j = 1, 2, . . . , m. If the norm | · | in the above theorem is Euclidean then all the functions in (9), (10) are polynomial. Hence, to verify conditions (9), (10) one can apply the technique of estimating the Lojasiewicz exponent mentioned above. As can be proved by standard reasoning [11,22] condition (10) is equivalent, in fact, to the following condition: there are numbers q, δ > 0 such that for small x and all y, which is similar to (7). Remark that the technique used in proving Theorem 1.3 is much the same that used in proving Theorem 1.2. Moreover, since both theorems, Theorem 1.2 and Theorem 1.3, provide necessary and sufficient conditions for sv-sufficiency of mapgerms under the same assumptions then condition (6) must be equivalent to (9) while condition (7) must be equivalent to (11). Nevertheless, no direct proofs of such an equivalence, to the best of the author's knowledge, are known.
The aim of the present paper is quite modest. First, we would like to reformulate the Kuo conditions (6), (7) in such a way to avoid verification of any inequalities in a horn-neighborhood of the a'priori unknown set f −1 (0). Second, we would like to replace the function D(·) in (6), (7) by something easier computable in applications.
To implement this program we firstly formulate in Section 2 "qualified" versions for the notions of regularity of the set of small non-zero solutions of equation (1) and transversality of this set to small spheres. The corresponding notions will play a key role in the further considerations. In Lemma 2.1 we show also that for polynomial maps these regularity and transversality conditions are equivalent to each other. Then, in Theorem 3.1 we formulate a set of equivalent to each other conditions (18), (19) for v-sufficiency (equiv. sv-sufficiency) of map-germs which are direct (and trivial) generalization of conditions (9), (11) from Theorem 1.3. Here we demonstrate also that these conditions may be treated as a natural generalization of both the Kuo conditions (6), (7) and the Thom conditions (5). At last, in Section 4 to prove Theorem 3.1 we establish equivalence between conditions (18), (19) and the Kuo conditions (6), (7).

2.
Qualified regularity and transversality. Before to start formulating main results of the paper, let us introduce some notions.
From now on ·, · stands for the Euclidean scalar product in R n , and | · | denotes the corresponding norm. If f : R n → R m is a smooth map then (df ) * (x) is the matrix conjugate to df (x). Clearly, the matrix (df ) * (x) consists of m column vectors grad f j (x), j = 1, 2, . . . , m.
Given a map-germ f : R n → R m with f (0) = 0 and an integer p ≥ 1 let us consider two auxiliary functions of variables x ∈ R n and y ∈ R m : and Note, that both the functions R p (f ; x, y) and T p (f ; x, y) are homogeneous in y. These functions are polynomials in x and y if f is a polynomial and p is even.
Positivity of the function R p (f ; x, y) for y = 0 and small x = 0 means that |(df ) * (x)y| > 0 for each y = 0 and all small non-vanishing solutions x of equation (1), that is the derivative of the map f (x) is regular on small solutions x of equation (1). So, the inequality R(f ; x, y) > 0 for x, y = 0 may be treated as a condition of regularity [8] of small non-zero solutions of equation (1). Therefore the relation valid for small x and all y with some C, q > 0, can be called the condition of qualified regularity of small non-zero solutions of equation (1). Similarly, positivity of the function T p (f ; x, y) for y = 0 and small x = 0 means that |(df ) * (x)y| · |x| > | (df ) * (x)y, x | for each y = 0 and all small non-vanishing solutions x of equation (1). The latter inequality is an algebraic representation of the fact that the set of small solutions of equation (1) is transversal to any small sphere |x| = ε [8]. Therefore the relation valid for small x and all y with some C, q > 0, can be called the condition of qualified transversality of small non-zero solutions of equation (1) to the small spheres |x| = ε.
If the map f is polynomial then the functions R p (f ; x, y) and T p (f ; x, y) are comparable, in a natural sense, for small x.
for small x and all y.
If a map f is polynomial then by Lemma 2.1 the set of small non-zero solutions of equation (1) is regular if and only if it is transversal to the small spheres |x| = ε, which is a well known fact [24]. In this case the set of small non-zero solutions of equation (1) is also qualifiedly regular (with some parameter q > 0) if and only if it is qualifiedly transversal (with the same parameter q) to the small spheres |x| = ε. Moreover, all conditions (14) and (15) with a given q > 0 but different p ∈ N are equivalent to each other.
for small x and all y where K is any one of the functions R p or T p .
as x → 0, x = 0, uniformly with respect to y = 0, where K is any one of the functions R p or T p .
Clearly, each function K (f (r) ; x, y) in Theorem 3.1 is a polynomial in x and y, homogeneous in y. This allows to simplify the formulation of Theorem 3.1 in the function case (m = 1). Set 1) if and only if for any p ∈ N there is a number q > 0 such that for small x where K * is any one of the functions R * p or T * p . For f ∈ E [r+1] (n, 1), the jet j r f (0) is v-sufficient (equiv. sv-sufficient) in E [r+1] (n, 1) if and only if for any p ∈ N K * (f (r) ; x) |x| pr+p → ∞ as x → 0, x = 0, where K * is any one of the functions R * p or T * p .

Remark 1.
Given an analytic map-germ h : R n → R 1 with h(0) = 0 and 0 < θ < 1, then the following Bochnak-Lojasiewicz inequality Hence for the polynomial f (r) (x) in Theorem 3.2 there is a number γ > 0 such that x) for small x. The latter inequalities mean that conditions (20) and (21) with K * = R * 1 are equivalent to the Kuiper-Kuo conditions (3) and (4), respectively. So, conditions (18) and (19) in Theorem 3.1 may be treated as a natural generalization of the Kuiper-Kuo conditions (3) and (4), respectively.

Remark 2. Direct verification shows that
and condition (20) with K * = T * 2 is nothing else than the Thom condition (5) for the map f (r) .
So, conditions (18) and (19) in Theorem 3.1 may be treated also as a natural generalization for the map case (m > 1) of the Thom condition (5).
As an example of application of the formulated above theorems, consider the well known problem on bifurcation of small auto-oscillations from an equilibrium in a system described by a differential equation with a parameter.
Therefore by Theorem 3.1 the jet j 2 f (0) is sv-sufficient in E [2] (4, 2). Then the set of small solutions of equations (23) consists of a pair of two-dimensional planes intersecting at the point τ = λ = ξ = η = 0. Existence of one of the planes of solutions of equations (23) is obvious, it is the plane ξ = η = 0 corresponding to the trivial periodic solution u(t) ≡ 0 of equation (22). Existence of the second plane of solutions of equations (23), passing the point τ = λ = ξ = η = 0 but different from the plane ξ = η = 0, testifies that equations (23) have nontrivial solutions with arbitrarily small τ = T − 2π, λ and {ξ, η} = 0. Hence, equation (22) has small nonzero periodic solutions for some arbitrarily small values of the parameter λ, see [14,16]. 4. Proofs. Throughout this section, O(t k ) with k ≥ 0 stands for values having an upper bound of the form c|t| k for small t with some c < ∞. Analogously, o(t k ) denotes values of a higher order of smallness than |t| k for small t.
Before starting to prove Theorem 3.1 let us prove first Lemma 2.1.

4.1.
Proof of Lemma 2.1. Inequalities (16) are a straightforward consequence of the following two-sided form of the power mean inequality So, we need only to prove inequalities (17) for a given p ∈ N. The right inequality in (17) is obvious. Therefore, it remains only to prove the left inequality in (17) which will be done by reductio ad absurdum.
If the left inequality in (17) is not valid then there are Since the functions T p (f ; x, y) and R p (f ; x, y) are homogeneous in y with the same power of homogeneity p ∈ N then without loss of generality one may suppose that |y i | = 1 and y i → y * , |y * | = 1. Let us write the following system of polynomial equalities and inequalities 1 : By the definition of the sequences {x i }, {y i } and {η i }, the set determined by the the relations (24) is not empty, and the point x = ϕ = ψ = η = u = v = w = 0, y = y * belongs to its closure. Hence, by the Curve Selection Lemma for semialgebraic sets (see, e.g., [24]) there are a number ε > 0 and real analytic around the origin functions x(t), y(t), ϕ(t), ψ(t) and η(t) satisfying the conditions or, what is the same by (12) and (13), |f (x(t))| p |y(t)| p + |(df ) * (x(t))y(t)| p |x(t)| p = R p (f ; x(t), y(t)) (25) and The latter relations imply from which by (25) Relations (26) imply also By dividing the both sides of the last inequality on |(df ) * (x(t))y(t)| p |x(t)| p , we obtain by (28) Because the functions x(t), y(t), ϕ(t), η(t) are real analytic for small t then they can be represented in the following form: Since f (x) is a polynomial and the functions x(t), y(t) are analytic then the functions (df ) * (x(t))y(t) and f (x(t)) are also analytic, and f (x(t)) → 0 as t → 0. Therefore by inequalities (27)-(29) there are integers k ≥ 1, l ≥ 0 such that Substituting now representations (31)-(34) for the related functions in (27) we get from which (since y * = 0) k ≥ r + s.

4.2.
Proof of Theorem 3.1. By Lemma 2.1 the conditions (18) for different p ∈ N and K = R p or K = T p are equivalent to each other, and the same is valid for the conditions (19). So, to prove Theorem 3.1 we need only to show that the Kuo condition (6) is equivalent to the condition (18) with K = R 1 : for small x and all y, while the Kuo condition (7) is equivalent to the condition (19) with K = R 1 : as x → 0, x = 0, uniformly with respect to y = 0. To prove equivalence between (6) and (40) introduce first, for a given set of vectors v 1 , v 2 , . . . , v m ∈ R n , the quantity D(v 1 , v 2 , . . . , v m ) as follows: where the minimum is taken over all m-tuples of real numbers y 1 , y 2 , . . . , y m satisfying m i=1 y 2 i = 1.
Represent now the vector (df (r) ) * (x)y in (40) in the form where y 1 , y 2 , . . . y m are the components of the vector y and f (r) m are the components of the map f (r) . Then, taking the minimum in the left-hand part of (40) over all the vectors y satisfying m i=1 y 2 i = 1, we obtain that Therefore (40), for small x, is equivalent to the condition: . . , grad f (r) m (x)) · |x| ≥ q|x| r . Then, taking into account that where D is the function (8), see [25, p. 348], we may state that (40), for small x, is equivalent also to the condition: with an appropriateq > 0. Now, let (6) be valid. Then for x ∈ H r (f (r) ; σ), |x| < ε, the first summand in the left-hand side of (45) is greater than σ|x| r . At the same time for x ∈ H r (f (r) ; σ), |x| < ε, by (6) the second summand in the left-hand side of (45) is greater than C|x| r . So, for |x| < ε, (6) implies (45) withq = min{σ, C}.
The proof of equivalence between (7) and (41) is a bit more complicated. First, to prove that (7) implies (41) we will show that (7) is not valid provided that (41) is not valid. To do it, we will need the following lemma the proof of which is relegated to Section 4.3 below.

4.3.
Proof of Lemma 4.1. Denote by H the class of polynomials in x of the form η(x) = x, v p x, w q u where p+q = r+1 and v, w ∈ R n . Since for such polynomials then, by the identity (dη) * (x)y, z ≡ y, dη(x)z , it is valid also the identity The last formula will be needed below in two cases: if η(x) = x, v r+1 u, and if η(x) = x, v r x, w u. First, let us construct a polynomial η i (x) ∈ H such that for the map g(x) = f (r) (x) + η i (x) the second inequality (46) be valid. Since f (r) does not satisfy (41) then by the Curve Selection Lemma [24] there are analytic around the origin functions for which Clearly, the function (df (r) ) * (x(t))y(t) is also analytic. If it is identically zero then it suffices to set η 1 (x) ≡ 0. In the opposite case let us represent it in the form (df (r) ) * (x(t))y(t) = zt γ + o(t γ ), z = 0, γ ≥ 1 is integer.