FUNCTIONAL DIFFERENTIAL EQUATIONS VOLUME 16 2009, NO 4 PP. 769- 779 ZERO SETS OF FUNCTIONS AND THEIR PIECEWISE-POLYNOMIAL APPROXIMATIONS * Y. YOMDIN t Abstract . Any closed set is a set of zeros of an infinitely differentiable function. On the other hand, for many important classes of functions f (polynomials, eigenvalues of elliptic operators, analytic functions with a controlled growth, smooth functions with a control of high-order derivatives, etc.) their zero sets Z(f) behave qualitatively like semialgebraic sets, and the "size" of Z (f) (in various senses) can be explicitly bounded. In this paper we relate the geometry of the zero set of a given function with the rate of its piecewise-polynomial approximation. 1. Introduction. It is well known that any closed set F C Rn is a set of zeros of an infinitely differentiable function f : Rn - t JR. On the other hand, for many important classes of functions f their zero sets Z(f) behave qualitatively like semi-algebraic sets, and the "size" of Z(f) (in various senses) can be explicitly bounded. Such classes include polynomials, eigenvalues of elliptic operators, analytic functions with a controlled growth, smooth functions with a control of high-order derivatives, etc. In particular, in [24] it was shown that assuming that f : BJ: - t JR (where BJ: is the unit ball in JRn) is a Ck-function, max if(x)l = 1 while the k-th derivative of f is sufficiently small, the zero set Z (!) of f behaves qualitatively like a semi-algebraic set, and its size can be explicitly bounded. Under roughly the same assumptions the bound on the volume of an €-neighborhood of Z(J) · · · ·····was··-obtained··-in···[2)-::············································-··· -·······················································································-·································-········ ··· ················ In the present paper we relate the geometry of the zero set of a given function with the rate of its piecewise-polynomial approximation. This relation may be considered as an additional manifestation of the following phenomenon (see [25, 26]): * To the memory of Professor Michael Drahlin. t Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel. 769 770 Y. YOMDIN Many important properties of functions, traditionally associated with their regularity, are in fact determined by usemi-algebraic complexity" of these functions, which is the rate of their semi-algebraic approximation. In the present paper we push the above observation to its extreme, assuming no regularity of the involved functions at all, besides their measurability. We show that the "size" of the zero set Z (f) of f : Bf -+ JR, in particular, the volume of this set and its <:-entropy are bounded in terms of the complexity of piecewise-polynomial <:-approximations of f. 2. Metric entropy of semi-algebraic sets. In this section we recall results of G. Vitushkin ([22, 23]) concerning metric entropy of semialgebraic sets. Since for our applications we need a rather accurate bounds on the "semi-algebraic constants" involved, we provide a simple proof of Vitushkin's theorem in one specific case. Let us recall a general definition of metric entropy. Let A C X be a relatively compact subset in a metric space X. DEFINITION 1. Fore> 0 the covering number M(e, A) is the minimal number of closed £-balls in X covering A. The binary logarithm of the covering number, H(<::,A) = logM(e,A) is called the €-entropy of A. The notion of "covering number" is very popular in fractal geometry and can be traced at least to Minkovski (see [17, 12]). See [15, 16] and many other publications for computation of €-entropy of various functional classes. Intuitively, <::-entropy of a set A is the minimal number of bits we need to memorize a specific element of the set A with the accuracy t::. In our setting X = lR11 • It will be convenient in this case to modify slightly the definition of the covering number, and to consider coverings by the e:-cubes Q~ which are translations of the standard £-cube Q~ = [0, e:] 11 • Q~ are the ~ - balls in the t=-norm. In particular, this choice implies that the leading t erm in the expression forM(£, A) has the form M(e:, A) = vo;£A) + .... A semi-algebraic set A in lR11 is the one obtained as a result of the standard set-theoretic operations over the sets defined by polynomial inequalities P(x1, ... , x11 ) ~ 0 or P(x 1 , ... , Xn) < 0. The diagram D(A) of A is the collection of the discrete data in the definition of A: it consists of the dimension n of the ambient space, of the degrees of the participating polynomials and of all the set-theoretic formulas applied. The following bound on the covering number of semi-algebraic sets was obtained by Vitushkin ([22, 23]; see also [13, 27]): PROPOSITION 1. Let A C Q~ be a semi-algebraic set with the diagram D. 771 ZERO SETS OF FUNCTIONS Then for any e > 0 we have (1) M(e, A) ~ Co(D) + C1(D)(?::) + · · · + Cn-l(D)(?:: )n-l + J.£n(A)(! )n. € € € Here J.£n(A) denotes the n-dimensional Lebesgue measure (or the volume) of the set A. The constants C0 , C1 , ... , Cn-1 , Cn depend only on D. The original proof was obtained in a much more general situation, via the ''Vitushkin variations" (see [22, 23, 13, 27]). We give here a detailed proof only in dimensions 1 and 2 and only for sets A being the sub-level sets of polynomials. Because of a natural scaling of the covering number we can assume r = 1. Let P(x) = P(x 1 , •.. , Xn) be a polynomial of degree d. We shall consider heresemi-algebraicsetsoftheform Wp(P) = {x E B}, P(x) ~ p}. In particular, we denote by Vp(P) = Wp2(P2 ) the sub-level set {x E Bi, !P(x) ! ~ p} of the polynomial P. The set Wp(P) is semi-algebraic, and it has a very special form: this set is defined by exactly one polynomial inequality P(x) ~ p. So the diagram D (Wp(P)) is completely determined by the pair (n,d). In this case the constants Ci (D) = Ci (n, d) can be given explicit ly. . Consider first the case n = 1. Here the set Wp(P) consists of at most [~] + 1 subintervals ~i in [0, 1] since its boundary { P(x ) = p} consists of at most d + 2 points (including the end points of [0, 1]) by the Bezout theorem. Let us cover each of these subintervals ~i by the adjacent €-intervals, starting with the left endpoint. Since all the adjacent £-intervals, except possibly one, are inside ~i their number does not exceed l~il(:) + 1. Altogether we have at most [~] + 1 + J.£1 (V,(P))(~) covering £-intervals. In two dimensions we consider the sub-level set W = Wp(P) inside the unit cube Qf = [0, 1]2 with P = P(x1 , x2 ) a polynomial of degree d. Let us subdivide Qi into adjacent €-cubes Q!i by vertical and horizontal straight lines. We have in total 2([~] + 1) straight segments. Mark for our covering all those Q!i which intersect W. Some of the marked Q!i are contained in W. The number of such boxes 1E ) 2 • Some other boxes Q!i cross the boundary 8W of does not exceed ~-t2 (V)( ,• ··~----· ..··- · ·w·:. ··To·eaclcsuclrQ?-we··crur·ru=rsuciate··(exclusively)--the·first--crossing-·point···---................... of its boundary with 8W (assuming that an orientat ion of oW was fixed) . Hence the number of the boxes of the second type does not exceed the total number of the crossing points of 8W with the partition segments. By the Bezout t heorem a straight line can meet 8W at not more than d points (since 8V = {P = p}). Altogether we have at most 2d( [ ~) + 1) ~ 4d(:) boxes of the second type. Finally, there may be covering boxes containing completely some of the connected components of W . Their number does not exceed (d - 1) 2 since 772 Y. YOMDIN inside each compact connected component of W there is a critical point of P. The last are defined by the system of two polynomial equations of degree d - 1 and the number of their solutions is bounded by (d - 1)2 via the two-dimensional Bezout theorem. (We can assume that all the solutions of this system are non-degenerate since in our proof the polynomial P can be perturbed slightly). Exactly the same approach can be extended to higher dimensions and to more complicated semi-algebraic sets. The technical details become more involved, but the general form of the bounds remains the same. In particular, for the sub-level sets W = Wp(P) and V = Vp(P) the constants Ci(n,d) have the form Ci(n, d) = Ci(n)d(n-i). Indeed, these constants appear, as in the proof for the dimension 2 given above, as the bounds on the number of the connected components of n - i - dimensional plane sections of our semialgebraic set, which have the required form (see, for example, [27]). This completes the proof of the Vitushkin theorem in our special case. Let us summarize the specific bounds obtained. PROPOSITION 2. For a sub-level set W = Wp(P) or V = Vp(P) inside the cube Q; in IR, or the cube Q; in IR 2 , we have, respectively, d M(c, W) ~ [2] 1 + 1 + Pl(W)(; ), 1 M(t, V)::; d + 1 + Pl(V)( ~ ), r 1 £ £ r 1 £ € M(t, W) ~ (d- 1) 2 + 4d(-) + p 2 (W)(- ?, M(t, V) ::; (2d- 1) 2 + 8d(-) + p 2 (V)(- ?. For a sub-level set W (V) inside the cube Q~ C IR"' we have n-1 M(t, W) ::; L Ci(n)d<n-i}(~Y + f.ln(W)( ~t, i= O E £ n-1 M(£, V) ::; L Ci(n) (2d)(n- i) (:c )i + f.ln(V)( ~ t. i=O £ £ ZERO SETS OF FUNCTIONS 773 3. Volume of sub-level sets of polynomials: Remez inequality. Let P(x) = P( x 1 , .•. , Xn) be a polynomial of degree d. In this section, to fit the traditional notations, we consider a part of the polynomial sub-level set inside the usual Euclidean ball. So we denote by V = Vp(P ) the sub-level set {x E Br, !P(x)! ~ p} of P. For n = 1 the volume of V can be estimated via the following Remez inequality ([21]): THEOREM 1. Let P(x) be a polynomial of degree d. Then max[-l,IJIP(x)! ~ Td( 4 - It), It where Jt = ~t 1 (Vi(P) and Td(x) = cos(darccos(x)) is the d-th Chebyshev polynomial. For other related bounds as well as for various multidimensional generalizations of the classical Remez inequality see, in particular, [3, 4, 5, 6, 11), [18]-.[20]. Specifically, we shall use in this paper the following Yu.BrudnyiGanzburg inequality ([6]), which is, probably, the most direct generalization of the classical Remez's one: THEOREM 2. Let l3 c !Rn be a convex body and let n C l3 be a measurable set. Then for a real polynomial P(x) = P(x1 , ... , xn) of degree d we have 1 (1) sup IPI ~ Td( B 1 + (1- .\):n 1 ) 1- (1- .\)n sup !Pl. n Here .\ = ~:~~~. This inequality is sharp and fo r n = 1 coincides with the classical Remez inequality. As a corollary (which is not sharp but sufficient for our applications) one has PROPOSITION 3. In the above notations (2) ~----------------- ----------~- ....--······---------~------····..·--..._..~--~--------------------········-·-------------- ··--------------------·-····----·~-·-······-·---··--·~-·-·--·~···~····~-··--- (3) Let us now assume that on the cube Q~ of the size r the SUPQ;: !PI = M. Then from Proposition 3 we get PROPOSITION 4. The volume of Vp(P) satisfies (4) 774 Y. YOMDIN 4. Tubular neighborhoods of sub-level sets. A 6-tabular neighborhood T0 (S) of a set S consists of all the points at the distance at most 8 from S. Metric entropy provides a convenient tool to study the geometry of T0 (S) because of the following simple lemma: LEMMA 1. M(2E, T~(S)) :5 M(E, 8). Proof: Indeed, if certain €-balls cover S, the 2E-balls centered at the same points cover ~ (S). Now combining Lemma 1 with Propositions 2 and 4 we get the following: THEOREM 3. For a polynomial P of degree dover the cube Q~ we have n-1 (1) M(2E, TdYp(P) n Q~])::; ~ Ci(n)d(n-i)(~)i + 4nrn(~)~(}t. In particular, for the volume of the tabular neighborhoods we get n-1 L 2nCi(n)d(n-i)riEn-i+2(n+2)nrn( ~)~ i=O n-1 = L Ci(n)d(n-i)ri€n-i+2(n+2)nrn( ~)~' i=O where Ci(n) = 2nCi(n). Assuming € :5 ~ and r :5 1 we get J.Ln(Te[Vp(P) n Q~]) ::; n-1 L Ci(n)ri(dc)n-i + 4nrn( ~)~ = i=O n-1 = d€ L Ci(n)ri(dE)"-i-l + 4nrn( ~)~ :=:; C(n)dE + 4nrn( ~)~ . i=O We summarize this computation in the following proposition: PROPOSITION 5. For a polynomial P of degree d over the cube E :5 ~ we have QJ. and for In particular, for p = 0, i. e. for the zero set Z(P) we get J.Ln(TdZ(P) n Q~]) :5 C(n)d€. This last inequality was obtained by different methods in [14, 1] (see also [2),[7]-[10]) in the course of study of tabular neighborhoods of the nodal sets of the Laplacian eigenfunctions. ZERO SETS OF FUNCTIONS 775 5. Piecewise-polynomial approximations. We consider measurable functions f : Q1 ~ R DEFINITION 2. A piecewise-polynomial £-approximation A off consists of 1. A covering of Q1 by a collection of sub-cubes Qi = Q~f, j = 1, ... , p, and 2. A collection of polynomials Pi of degree di , such that for any j = 1, ... ,p we have supQ~~ If- Pi I : : :; c 3 5.1. Volume of the zero set of f. Now we define an invariant of a piecewise-polynomial <:-approximation A of f which will bound the volume of the zero set of f: DEFINITION 3. Let 6 2:: 0 be given. We define v;-:(6, A ) as (1) p & i=I Mi 1 v~(o,A) =4n I :rj(- ) d;, where Mi = supQ~i JPil· :J Now applying the inequality of Proposition 4 .we get one of our main results: THEOREM 4. The volume of the zero set Z(f) = {x E Q].\ f (x ) = 0} satisfies (2) where A runs over all the piecewise-polynomial <:-approximations off. Proof: Over each sub-cube Q~! the zero set Z(f) off is contained in the sub' level set ~(Pj), since supQ~i lf-Pil :::::; e by definition of the£ approximations. :J The result now follows from Proposition 4 applied to each of the polynomials Pi of degree di on the sub-cube Q~f. Remark Using in the definition of v:(&, A) instead of Proposition 4 a sharp ...,-~----- --beund-fer·-J:Ln·(-~f.P))RQ?-j--whiGh-Gan--be--obtained..from..Theorem..3.2..we ..obtain........................... .. - a more accurate invariant v;-:(o, A). The corresponding version of Theorem 5.1 is given by the inequality (3) where A runs over all the piecewise-polynomial e-appro:ximations of f . As we shall see below, inequality (3) is sharp, while Theorem 5.1 is sharp only up to a constant C(n). 776 Y. YOMDIN 5.2. Metric entropy of the zero set of f. Our next result is the bound on the metric entropy of Z(f) similar to the bound on the volume provided by Theorem 5.1. We need the following definition: DEFINITION 4. For a piecewise-polynomial f.-approximation A off the "i-th variation" vf(A) fori = 0, 1, ... , n- 1 is defined as p (4) vin(A) = ~ C ( )d(n-i) rJ, i ~ i n J j=l where Ci(n) are the constants defined in Proposition 2.2. The following theorem is the second main result of this paper: THEOREM 5. The covering number M(E, Z(f)) of the zero set Z(f) = {x Q1, f(x) = 0} satisfies E (5) where A runs over all the piecewise-polynomial f.-approximations off. Proof: Over each sub-cube Q~J the zero set Z(f) off is contained in the sub' level set ~(PJ), since supQ~j If -.Pjl :::; e by definition of thee approximations. ' Applying Proposition 2 to each of the polynomials PJ of degree d5 on the subcubes Q~J1 we have n- 1 M(t,Z(f)nQ~f) :5 M(e, Ve(Pj)nQ;f)::; LCi(n)dJn-i)(:)i+JLn(~(PJ))(~ti=O ..l. Now by Proposition 4 we have Jln(~(PJ))) :::; 4nrj(~) dj. Summing up for j = 1, ... ,p and using Definitions 2 and 4 we get the required result. 6. Some examples. In this section we construct examples of functions f for which the rate of piecewise-polynomial approximation (versus the complexity of the approximant) may be quite arbitrary while the bounds of Theorems 5.1 and 5.2 are essentially sharp. Let Q1 = Ufr= 1 Q~j be a non-overlapping partition of the cube Qf. For each j = 1, 2, ... we fix a polynomial PJ(x) = PJ(xb . .. ,xn) of degree dj· Now we define a piecewisepolynomial function g(x) by 00 (6) g(x) = L XQ;:jPJ(x), j=l ) 777 ZERO SETS OF FUNCTIONS where XQ~? is the characteristic function of the cube Q~j. Now let 3 given. We define 9e(x) by (7) 9e(x) = g(x) , !g(x)l 2 €, € > 0 be 9e(x) = 0, !g(x)l < €. Clearly, lge(x) - g(x) l ::; € and hence according to Definition 2 the partition Qf = U~ 1 Q~j and the function g define a piecewise-polynomial €approximation A of gE. For the zero set Z(ge) we have Z(gE) = U~ 1 Ve(Pj) n Q~j and this union is disjoint. Hence J.Ln(Z(ge)) = 1 J.Ln(Ve(Pi) n Q~j). Now since the Remez inequality in Theorem 3.1 is sharp (and its version given by Propositions 3, 4 is sharp up to constants C(n)) we can choose the polynomials Pi in such a way that for each j = 1, 2, ... the volume of the set "Ve(Pj) on Qr:j is given, up to a constant C(n), by the right hand side of the expression (4) of Proposition 4. Therefore we have 2:; 00 f.tn(Z(ge)) = LJ.Ln(Ve(Pj) n Q;f) 2 j =l where v;: (€, A) is given by Definition· 3. This inequality shows that for the function 9e the upper bound of Theorem 5.1 is sharp, up to a constant C(n ). Notice that the complexity of the piecewise-polynomial function g depends on the choice of the covering Q";;i and of the degrees dj of the polynomials Pi. Exactly in the same way we can show that the metric entropy bound provided by Theorem 5.2 is sharp up to a constant C(n) . The only difference is that we have to take into account that for each specific € > 0 some of the cubes Q~!3 can be entirely covered by one €-ball. However, if we consider only -,-----finite_padition!Lan<Ltake~umfficie.ntly_s_mJill, the §h~gpness of the bound m =·::=---~ Theorem 5.2 can be shown exactly as above. Remark. It would be important to investigate the asymptotic behavior of the bounds in Theorems 5.1 and 5.2 as € tends to zero, at least for some natural classes of the approximated functions f. However, while the control of the approximation accuracy is provided by classical results of Approximation Theory, the control of the constants Mi turns to be a rather subtle problem. It is somewhat similar to the problems treated in [2, 24j and we plan to present some results in this direction separately. 778 Y. YOMDIN REFERENCES [lj P. Balint, N. Chernov, D. Szasz, L P. Toth, Geometry of multidimensional dispersing billiards, Asterisque, 286 (2003), 119-150. [2] P. Batchourine, Oh. Fefferma.n, The volume near the zeros of a smooth function. Rev. Mat. Iberoam. 23 (2007), no. 1, 259-267. [3] A. Borichev, F. Nazarov, M. 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Comte, Tame Geometry with Applications in Smooth Analysis, Lecture Notes in Mathematics, 1834, Springer, Berlin, Heidelberg, New York, 2004. -······~····¥••··· ···· ·--------- ----- -· ·· ······························ ·· -- - ---- · · -------·········-············------ ----- ------ ------ ··············~·-····----------- -- - -- - -- -····························· --- ---- -------- -----· · ······ ·· ········-······¥······· · ·· ·· · --- - ----.....•..•.•..••......•.....••..•.•.........•....•....••...........•...•....•... (