ON A REAL ANALOGUE OF BEZOUT INEQUALITY AND THE NUMBER OF CONNECTED COMPONENTS OF SIGN CONDITIONS

Transcription

1 ON A REAL ANALOGUE OF BEZOUT INEQUALITY AND THE NUMBER OF CONNECTED COMPONENTS OF SIGN CONDITIONS SAL BARONE AND SAUGATA BASU Abstract Let R be a real closed field and Q 1,, Q l R[X 1,, X k ] such that for each i, 1 i l, deg(q i ) d i For 1 i l, denote by Q i = {Q 1,, Q i }, V i the real variety defined by Q i, and k i an upper bound on the real dimension of V i (by convention V 0 = R k and k 0 = k) Suppose also that 2 d 1 d 2 1 k + 1 d 3 1 (k + 1) 2 d 4 1 (k + 1) l 3 d l 1 1 (k + 1) l 2 d l, and that l k We prove that the number of semi-algebraically connected components of V l is bounded by O(k) 2k d k j 1 k j j d k l 1 l 1 j<l This bound can be seen as a weak extension of the classical Bezout inequality (which holds only over algebraically closed fields and is false over real closed fields) to varieties defined over real closed fields Additionally, if P R[X 1,, X k ] is a finite family of polynomials with deg(p ) d for all P P, card P = s, and d l 1 d, we prove that the k+1 number of semi-algebraically connected components of the realizations of all realizable sign conditions of the family P restricted to V l is bounded by O(k) 2k (sd) k l 1 j l d k j 1 k j j These results have found applications in discrete geometry, for proving incidence bounds [11], as well as in effcient range-searching [20] Contents 1 Introduction 2 11 History and motivation 2 12 Applications in discrete geometry 4 13 Failure of the naive version of Bezout inequality over the reals 5 14 Main results 6 15 Outline of the proofs of the main theorems 8 2 Preliminary results Deformation of several equations to general position Mathematics Subject Classification Primary 14Q20; Secondary 14P05, 52C10, 13D40 Key words and phrases real algebraic varieties, semi-algebraic sets, connected components, sign conditions The second author was partially supported by NSF grants CCF , CCF and DMS

2 2 BARONE AND BASU 22 Generic coordinates 15 3 Proofs of the main theorems Definitions and main properties of approximating semi-algebraic sets Bounds on the Betti numbers of non-singular complete intersections Proof of Theorem Proof of Theorem 5 31 References 32 1 Introduction 11 History and motivation Let R be a fixed real closed field, and we denote by C the algebraic closure of R Bounds on the number of semi-algebraically connected components, and in fact on all the Betti numbers of real algebraic varieties and of semi-algebraic subsets of R k in terms of the number and the degrees of the polynomials used to define them is a well studied problem in quantitative real algebraic geometry The classical bounds, going back to the work of Oleĭnik and Petrovskiĭ [23], Thom [27] and Milnor [21], bounded the sum of the Betti numbers of real algebraic varieties, as well as those of basic closed semi-algebraic sets These and related bounds (see below) are extremely important in real algebraic geometry [10], but have also been used extensively in other areas such as combinatorics [3], discrete and computational geometry [15], and theoretical computer science [22] (the cited references are not by any means exhaustive but only given for illustrative purposes we refer the reader to [9] for a more extensive survey) An important application of the bounds mentioned above is in bounding the number of semi-algebraically connected components of the realizations of various sign conditions of a family of polynomials in R k or more generally sign conditions restricted to a given real sub-variety of R k In order to state these results more precisely, we introduce some notation Notation 11 For P R[X 1,, X k ] a finite family of polynomials, a sign condition σ on P is an element of {0, 1, 1} P The realization Reali(σ, V ) of the sign condition σ on a semi-algebraic set V R k is the semi-algebraic set defined by Reali(σ, V ) = {x V sign(p ) = σ(p ), P P} Notation 12 For any finite family of polynomials Q R[X 1,, X k ] we will denote by Zer(Q, R k ) the set of real zeros of Q in R k If Q = {Q}, then we will use the notation Zer(Q, R k ) instead We will denote by Q h (respectively, Q h ) the homogenizations of the polynomials in Q (respectively, the polynomial Q), and denote by Zer(Q h, P k C ) (respectively, Zer(Qh, P k C )) the common zeros of the family Q h (respectively, the polynomial Q h ) in the projective space P k C Notation 13 For any Q R[X 1,, X k ] we will denote by deg(q) the degree of Q More, generally for a tuple of polynomials Q = (Q 1,, Q l ) R[X 1,, X k ] l we will denote deg(q) = (d 1,, d l ) where d i = deg(q i ), 1 i l Notation 14 For any semi-algebraic set S R k, we will denote by b i (S) the i-th Betti number of S In particular, b 0 (S) is the number of semi-algebraically connected components of S

3 ON A REAL ANALOGUE OF BEZOUT INEQUALITY 3 Notation 15 For any semi-algebraic set S R k, we will denote by dim S the real dimension of S For any x S, we denote by dim x S the local real dimension of S at x Note that unlike complex varieties, an irreducible real variety can have have different local dimensions at different points Remark 16 We will at times slightly abuse notation and use the same letter to denote a tuple of polynomials as well as the ordered finite set whose elements are the elements of the tuple This should not cause any confusion The following theorem gives a reasonably tight bound on the number of semialgebraically connected components of the realizations of all realizable sign conditions of a finite family of polynomials restricted to a variety It generalizes earlier results of Alon [3], Warren [29] and Pollack and Roy [24], and has found several applications in discrete geometry Theorem 1 [5] Let P, Q R[X 1,, X k ] be finite families of polynomials such that the degrees of the polynomials in P, Q are bounded by d, card P = s, and dim R (V ) = k, where V = Zer(Q, R k ) Then, b 0 (Reali(σ, V )) O(1) k s k d k σ {0,1, 1} P Notice that in the bound in Theorem 1, while the exponent of s depends on the dimension of the variety V, the exponent of d is that of the ambient space Moreover, the bound depends only on the maximum degree of the polynomials in P and Q This is a consequence of the fact that the proof involves taking sums of squares of the polynomials in P and Q, and thus only the maximum degree plays a role in the argument This feature of taking the sum of squares is something that is common in the proofs of all the bounds mentioned above As such they all depend on the maximum of the degrees of the polynomials used to define the given set or sign conditions More recently, a new application of the bounds described above in discrete and computational geometry, triggered by the work of Guth and Katz [16], raised the question whether even the part of the bound in Theorem 1 that depends only on the degree d could have a finer dependence on the degrees of the polynomials in P and Q, in the case when the degrees of the polynomials in Q and those in P differ significantly (see [16, 26, 18, 17, 30, 20]) This is one of the primary motivations behind the results proved in the current paper (see Section 12 below for more detail) A second motivation is to prove a version of the Bezout inequality on bounding the number of isolated complex solutions (or more generally the number of connected components) of an affine polynomial system by the product of the degrees, over real closed fields where the original statement of the inequality does not hold (see Section 13, and in particular Example 18 and Remark 111 below) A first step was taken in this direction in [4] where the authors of the current paper proved the following theorem (actually a more precise statement appears in [4] but the following simplified version is what is important for the present purpose) Theorem 2 [4] Let P, Q R[X 1,, X k ] be finite subsets of polynomials such that deg(q) d 1 for all Q Q, deg(p ) d 2 for all P P Suppose also that d 1 d 2, and the real dimension of V = Zer(Q, R k ) is k 1 k, and that card P = s

4 4 BARONE AND BASU Then, σ {0,1, 1} P b 0 (Reali(σ, V )) O(1) k (sd 2 ) k1 d k k1 1 Remark 17 One should compare Theorem 2 with Theorem 1 The new aspect of Theorem 2 is the more refined dependence on the two different degrees, taking into account the dimension of the variety V and the fact that d 1 d 2 Notice that Theorem 2 implies the following corollary about the number of semialgebraically connected components of real varieties Corollary 3 Let Q 1, Q 2 R[X 1,, X k ] such that deg(q 1 ) d 1,deg(Q 2 ) d 2, d 1 d 2 Let V 1 = Zer(Q 1, R k ) and dim V 1 k 1, and let V 2 = Zer({Q 1, Q 2 }, R k ) Then, b 0 (V 2 ) O(1) k d k k1 1 d k1 2 Proof In Theorem 2, take Q = {Q 1 }, P = {Q 2 } and V = V 1 12 Applications in discrete geometry While Theorem 2 (in particular, also Corollary 3) has already proved useful in certain applications in discrete and computational geometry (see [2, 26, 20]), some even more recent developments seem to require a more detailed analysis The requirement of refined bounds from real algebraic geometry in the applications mentioned above originates in the so called polynomial partitioning method due to Guth and Katz [16], which provides a framework for proving bounds in several types of problems in discrete geometry involving finite sets of points (such as incidence problems [26], unit and distinct distance problems [16, 17, 30] etc) The original polynomial partitioning result states that given any set, S, of n points in R k, and an auxiliary parameter r, 0 < r < n, there exists a polynomial P R[X 1,, X k ] of degree at most O ( r 1 k ), having the property that each semi-algebraically connected component C of R k \ Zer(P, R k ) contains at most n r of the points of S The number of such semi-algebraically connected components C (using for instance Theorem 1) is bounded by O(r), and it is at this point that a quantitative bound on the number of semi-algebraically connected components of a semi-algebraic set or sign conditions enters the proof The polynomial partitioning theorem is a tool to decompose a given problem involving the set S into sub-problems of smaller size (corresponding to the point sets C S where C is a semi-algebraically connected component of R k \ Zer(P, R k )) However, it might happen that most or even all the points of S are contained in Zer(P, R k ) which is problematic for a divide-and-conquer type argument In this case, an obvious idea is to try to extend the polynomial partitioning theorem to varieties of lower dimensions, and continue the partitioning recursively However, in order to prove the strongest result possible using this approach, one requires tight bounds on the number of semi-algebraically connected components of real varieties defined by a sequence polynomials of strictly increasing degrees, which has a much more refined dependence on the sequence of degrees than what was provided in Theorem 2 mentioned above (where the length of the sequence is restricted to at most 2) Note however that in the applications related to the polynomial partitioning method, the length of the sequence of degrees could be as large as the dimension of the ambient space, and Theorem 2 is insufficient to deal with cases with degree sequences

5 ON A REAL ANALOGUE OF BEZOUT INEQUALITY 5 of lengths greater than 2 The main result of this paper (Theorem 4) is geared towards handling this situation, and has already proved useful in applications involving the polynomial partitioning technique For example, Theorem 4 plays a crucial role in a recent application of multi-level polynomial partition technique for proving the tightest known bound for the point-hypersurface incidence problem in R 4 [11, Theorem 15] 13 Failure of the naive version of Bezout inequality over the reals Before stating our results let us consider what kind of refined bounds are plausible In the case of a real variety V of R k, which is a non-singular complete intersection (even at infinity) and defined by polynomials of degrees d 1 d 2 d l, the number of semi-algebraically connected components of V is bounded by (see Proposition 322 as well as Remark 323 below) O(1) k d 1 d l 1 d l d k l l Notice that k l = dim V It is thus natural to hope that such a bound continues to hold even if the given variety is not a non-singular complete intersection namely, one might hope that the number of semi-algebraically connected components of a real variety V R k defined by a sequence of l polynomials having degrees d 1 d 2 d l is bounded by O(1) k d 1 d l d dim l V However, the following well known (counter-)example (that appears in [14]) already shows that this is not the case Example 18 Let k = 3 and let Q 1 = X 3, Q 2 = X 3, 2 d Q 3 = (X i j) 2 i=1 j=1 The real variety defined by {Q 1, Q 2, Q 3 } is 0-dimensional, and has d 2 isolated (in R 3 ) points, whereas the degree sequence (d 1, d 2, d 3 ) = (1, 1, 2d), and thus the conjectured bound is d 1 d l d dim l V = O(d) In particular, this example shows that the (naive version of) Bezout inequality which states that the number of isolated complex zeros of a system of polynomial equations is bounded by the product of the degrees of the polynomials appearing in the system, is not true over if we replace the complex numbers by a real closed field Notice however that the polynomials Q 1, Q 2, Q 3 do not define a non-singular complete intersection While this might seem discouraging at first glance, one way to repair the situation is to formulate a bound that depends not just on the degree sequence and the dimension of the last variety V = V 3 = Zer({Q 1, Q 2, Q 3 }, R k ), but also takes into account the dimensions of the intermediate varieties V 1 = Zer(Q 1, R k ), V 2 = Zer({Q 1, Q 2 }, R k ) etc Notice that in Example 18 the dimensions k 1 = dim V 1, and k 2 = dim V 2 are both equal to 2, whereas k 3 = dim V 3 = 0 The number of semialgebraically connected components in this case is bounded by O(d k k1 1 d k1 k2 2 d k2 3 ), where d i = deg Q i This is the starting point of the formulation of the new results proved in this paper We prove the following theorems where the shapes of the bounds should be seen in the light of Example 18

6 6 BARONE AND BASU 14 Main results For the rest of the paper we will use the following notation Notation 19 Let Q 1,, Q l R[X 1,, X k ] such that for each i, 1 i l, deg(q i ) d i For 1 i l, denote by Q i = {Q 1,, Q i }, V i = Zer(Q i, R k ), and dim R (V i ) k i We set V 0 = R k, and adopt the convention that k i = k for i 0 It is clear that k = k 0 k 1 k l With these assumptions we have the following generalization of Corollary 3 Theorem 4 Suppose that Then, 2 d 1 d 2 1 k + 1 d 3 b 0 (V l ) O(1) k τ=(τ 0,τ 1,,τ l 1 ) 1 (k + 1) 2 d 4 F (k, τ) d τ l 1 l 1 j<l 1 i<l 1 (k + 1) l 2 d l ((k τ i 1 + 1)d i ) τi 1 τi where the sum on the right hand side is taken over all τ N l, with k = τ 0 τ 1 τ l 1 0, and τ i k i, for each i, 1 i < l, and ( ) k τ l 1 F (k, τ) = (k τ l 1 + 1) τ 0 τ 1, τ 1 τ 2,, τ l 2 τ l 1 This implies that b 0 (V l ) O(1) l O(k) 2k d kj 1 kj j d k l 1 l, and in particular if l k, b 0 (V l ) O(k) 2k 1 j<l d kj 1 kj j d k l 1 l Remark 110 Note that since the real dimension of each variety V i is at most the complex dimension of V i, Theorem 4 remains true if we replace real dimension by complex dimension in the statement This observation is important in the application of Theorem 4 to incidence problems (see [11]) Remark 111 In view of Example 18 above, Theorem 4 can be viewed as a weak version of the Bezout inequality over real closed fields The following slight modification of Example 18 shows that the dependence on the degrees in the bound in Theorem 4 cannot be improved Example 112 Let k = k 0 k 1 k 2 k l = 0, and d 1,, d l be even For 1 i l, let 2 k k i d i/2 Q i = (X j h) j=k k i 1 +1 Then, for 0 i l, deg(q i ) = d i, the real dimension of the variety V i = Zer(Q i, R k ) where Q i = {Q 1,, Q i }, is clearly k i, and b 0 (V l ) = 1 2 k dk0 k1 1 d k1 k2 h=1 2 d k l 2 k l 1 l 1 d k l 1 l

7 ON A REAL ANALOGUE OF BEZOUT INEQUALITY 7 With the same assumptions as in Theorem 4, suppose additionally that P R[X 1,, X k ] is a finite family of polynomials with deg(p ) d for all P P, and card P = s, and suppose that d l 1 k+1 d Theorem 5 (1) where is defined by = σ {0,1, 1} P b 0 (Reali(σ, V l )) τ=(τ 0,τ 1,,τ l ) 1 i l k l j=0 4 j ( s j ) O(1) k F (k, τ)d τ l ((k τ i 1 + 1)d i ) τi 1 τi, where the sum is taken over all τ N l+1, with k = τ 0 τ 1 τ l 0, and τ i k i, for each i, 1 i l, and ( ) k τ l F (k, τ) = (k τ l + 1) τ 0 τ 1, τ 1 τ 2,, τ l 1 τ l This implies that b 0 (Reali(σ, V l )) O(1) l O(k) 2k (sd) kl d kj 1 kj j σ {0,1, 1} P 1 j l In particular, if l k, b 0 (Reali(σ, V l )) σ {0,1, 1} P O(k) 2k (sd) kl 1 j l d kj 1 kj j Remark 113 Notice that in the case l = 1, the bound (1) in Theorem 5 implies that of Theorem 2, and hence Theorem 5 is a strict generalization of Theorem 2 With the same assumptions as in Theorem 5, let for P P, d P = deg(p ), and for any subset I P let ( ) d I = (k + 1) (card I 2 )+(k l card I)(card I 1) (2) d P (max d P ) k l card I P I We have the following variant of Theorem 5 (the extra precision with respect to the degrees of polynomials in P might be useful in applications in incidence geometry) Using Notation 19 and notation introduced in (2) above: Theorem 6 σ {0,1, 1} P b 0 (Reali(σ, V l )) 4 I P j=card I k l P I j O(1) l O(k) 2k d I 1 j l d kj 1 kj j Remark 114 The condition on the degrees in Theorems 4 and 5 might look unnatural at first glance but is forced on us by the method of the proof, which involves taking minors of matrices of size at most (k + 1) (k + 1) with entries which are polynomials of degree d i, 1 i l We want at each step, the degree d i to majorize the degree of the polynomial obtained as a minor in the previous step whose entries have degree at most d j, where j < i Notice that in the case l = 2,

8 8 BARONE AND BASU the condition on the degree sequence is just d 1 d 2, and this allows us to recover Theorem 2 from Theorem 5 Remark 115 We also note that in [25] the authors define the complexification of a semi-algebraic set as the smallest complex variety containing it, and prove an effective bound on the geometric degree of this complexification which depend amongst other quantities on the real dimension of the given set This degree could be thought of as the real degree of the semi-algebraic set It is possible that Theorem 5 could serve as an alternative basis for a good definition of the real degree of a real variety in the sense that the real degree of a real variety V should control the number of semi-algebraically connected components of the intersection of V with any real hypersurface of sufficiently large degree We do not pursue this idea further in the current paper Finally, we conjecture that the bounds in Theorems 4 and 5 extend to the sum of all the Betti numbers (instead of just the zero-th one) The techniques developed in this paper are not sufficient to prove this conjecture 15 Outline of the proofs of the main theorems The main difficulty that one faces in order to prove bounds having the shapes of Theorems 4 and Theorem 5 is that in order to respect the degree sequence one has to be careful about taking sums of squares which spoil the dependence on the degrees The crucial idea is to use the notion of approximating varieties An approximating variety is a variety which is infinitesimally close to the given variety of the same dimension, but having good algebraic properties which allow one to give a precise bound on the number of its semi-algebraically connected components in terms of the sequence of degrees of polynomials defining it (rather than just the maximum degree) If the given variety can be covered (in a technical sense made precise later) by a small number of such approximating varieties, then the problem of bounding the number of semi-algebraically connected components of the given varieties reduces to the problem of bounding the total number of semi-algebraically connected components of these approximating varieties The idea of using approximating varieties originates in algorithmic semi-algebraic geometry and it was used in [7] to give efficient algorithms for computing sample points on varieties and in [8] to compute roadmaps of semi-algebraic sets The combinatorial part of the complexities of these algorithms depends on the dimension of the given variety rather than that of the ambient space, and this is where the approximating varieties play an important role in those papers In quantitative semi-algebraic geometry, the notion of approximating varieties was used in [4] in order to prove Theorem 2 The approximation scheme that we use, which is a generalization of the one used in [4] is described in Section 31 below One difficulty in generalizing the scheme in [4] is that the non-singularity of polar varieties of smooth hypersurfaces with respect to generic projections that is used in that paper no longer holds for smooth varieties of higher co-dimension A second difficulty is that the sequence of local (real) dimensions at a point x V l of the varieties V 1,, V l is not globally constant, but is only a local invariant (see Example 33 and Figure 1) Thus, one cannot expect to have a single global approximating variety with good properties We overcome the latter problem by taking into account all possible sequences of local dimensions whether they actually occur or not (indexed by the set A below), and construct

9 ON A REAL ANALOGUE OF BEZOUT INEQUALITY 9 approximating varieties with acceptable degree sequences to approximate each of them Consider the subset of points of U i of V l having local dimension i k l At each point x U i the dimension of V l 1 is between i and k l 1 Suppose we have already constructed approximations of subsets of V l 1 consisting points having some fixed local dimension at V l 1 Using these approximations and adding appropriately many equations in each case we construct a set of approximations of U i Taking all these approximating varieties, for all i, 0 i k l, and noticing that V l is the union of the U i s we obtain a global approximation of V l (see Example 317 and Figures 3, 4, 5, 6 and 7 below) More precisely, we construct a family of basic semi-algebraic sets each of the form, Bas(P, Q) := {x R k P (x) = 0, Q(x) 0, P P, Q Q}, where R is some real closed extension of R depending on the particular approximating set The family of pairs {(P α τ,l, Qα τ,l )} τ A N l,α I(τ) defining these approximating varieties are indexed by a pair of indices τ, α coming from two finite set of indices A l N l, and I l (τ) While the definition of the second, I l (τ), is a bit technical and which we defer for later, the definition of the index set A l is the following A l := {τ = (τ 1,, τ l ) N l k τ 1 τ 2 τ l, τ i k i, 1 i < l} For any given τ A l, let V τ V l denote the closure of the set of points x V l such that the local real dimension of V i at x is equal to τ i, for each i, 1 i l The union of the approximating sets Vσ,l α = Bas(Pα σ, Q α σ) with σ τ, approximates V τ in a certain precise sense (see Proposition 313 below), and since clearly V l = τ A V τ, the union of all the approximating sets {Vτ,l α } τ A N l,α I(τ) approximate the whole variety V l Because of the approximating property, in order to bound the number of semi-algebraically connected components of V l it suffices to bound the sum of the number of semi-connected components of each one of the approximating sets Vτ,l α The tuples Pα τ,l, Qα τ,l have the following properties that enable us to obtain good bounds on the number of semi-algebraically connected components of Vτ,l α (see Proposition 213 below) a) The tuple of polynomials Pτ,l α define a non-singular, bounded complete intersection of dimension τ l k l In particular, this means that the cardinality of Pτ,l α is equal to k τ l Suppose that Pτ,l α = (P 1,, P k τl ) Let for 1 i l, l i = τ i 1 τ i, with the convention that τ 0 = k, and L i = i h=1 l h Then for each i, 1 i l, the degrees of the polynomials P Li 1+1,, P Li are bounded by O(kd i ) b) Q α τ,l is either empty or contains one polynomial, Qα τ,l, with deg(qα τ,l ) = O(d l), and P, Q α τ,l, where P is any subset of Pτ,l α, defines a non-singular complete intersection It remains to bound the number of semi-algebraically connected components of each Vτ,l α and take the sum of these bounds, for which we use the same result as in [4] where a bound is derived using a classical formula for the Betti numbers of complex non-singular complete intersections and the Smith inequality (see Proposition 322 below) The number of approximating varieties (which is independent of the given degree sequence) and the bounds on the degree sequences of their defining

10 10 BARONE AND BASU polynomials as stated in Properties a) and b) above are good enough to give us the bound in Theorem 4 Theorem 5 follows from Theorem 4 using standard techniques already used in [6] and no fundamentally new ingredients The rest of the paper is organized as follows In Section 2, we recall some basic facts about real closed fields of Puiseux series that we need for making deformation arguments We also recall some results proved in [4] on the choice of generic coordinates Finally, in Section 3 we prove the main theorems 2 Preliminary results 21 Deformation of several equations to general position In this section we describe how to deform a system of equations using infinitesimals so that the set of common zeros of the deformed equations (in certain real closed non-archimedean extensions of the ground field) has good properties For this we first need to recall some properties of Puiseux series with coefficients in a real closed field We refer the reader to [10] for further detail We begin with some notation Notation 21 For R a real closed field we denote by R ε the real closed field of algebraic Puiseux series in ε with coefficients in R We use the notation R ε 1,, ε m to denote the real closed field R ε 1 ε 2 ε m Note that in the unique ordering of the field R ε 1,, ε m, 0 < ε m ε m 1 ε 1 1 Also, note that both fields R ε, R δ are sub-fields in a natural way of R ε, δ Notation 22 If R is a real closed extension of a real closed field R, and S R k is a semi-algebraic set defined by a first-order formula with coefficients in R, then we will denote by Ext(S, R ) R k the semi-algebraic subset of R k defined by the same formula It is well-known that Ext(S, R ) does not depend on the choice of the formula defining S [10] Notation 23 For x R k and r R, r > 0, we will denote by B k (x, r) the open Euclidean ball centered at x of radius r If R is a real closed extension of the real closed field R and when the context is clear, we will continue to denote by B k (x, r) the extension Ext(B k (x, r), R ) This should not cause any confusion Notation 24 For elements x R ε which are bounded over R we denote by lim ε x to be the image in R under the usual map that sets ε to 0 in the Puiseux series x Notation 25 Let Q R[X 1,, X k ], 0 q k, and H R[X q+1,, X k ] Let ζ be a new variable We denote Def(Q, ζ, q, H) = (1 ζ)q ζh Notation 26 For P = (P 1,, P m ), with each P i R[X 1,, X k ], 1 q k, and G = (G 1,, G m ) with each G i R[X q+1,, X k ], and ζ a new variable, we denote by Def(P, ζ, q, G) the tuple (Def(P 1, ζ, q, G 1 ),, Def(P m, ζ, q, G m )), and by Def(P, ζ, q, G) h the corresponding tuple of homogenized polynomials (Def(P 1, ζ, q, G 1 ) h,, Def(P m, ζ, q, G m ) h )

11 ON A REAL ANALOGUE OF BEZOUT INEQUALITY 11 Notation 27 For F = (F 1,, F k p ), q p k, we denote the jacobian matrix F 1 X q+1 Jac(F, p, q) := F 1 X k F k p X q+1 F k p X k whose rows are indexed by [q + 1, k] and columns by [1, k p] For J [q + 1, k], card J = k p and k J, let Jac J denote the (k p) (k p) matrix extracted from the matrix Jac(F, p, q) by extracting the rows whose index are in J, and let jac J = det(jac J ) Let (3) F J := F {jac J {i}\{k} }, and the finite constructible set (4) i [q+1,k]\j C J (F) := {x Zer(F J, R k ) jac J (x) 0} Proposition 28 Let F = (F 1,, F k p ), each F i R[X 1,, X k ], and such that the variety Zer(F h, P k C ) is a non-singular complete intersection Let x Rk be a non-generate critical point of the projection map to the X k -coordinate restricted to the variety V = Zer(F, R k ) Then, there exists a subset J [1, k], card J = k p, k J, satisfying the following two conditions 1 The (k p) (k p) matrix, Jac J, extracted from the matrix Jac(F, p, 0) by extracting the rows whose index are in J, evaluated at x is non-singular 2 The point x is a simple zero of the system F J (see (3) for definition) Proof First note that using the Jacobian criteria for non-singularity of real algebraic varieties (see for example [12, Definition 334]), we have that the variety V is of dimension equal to p and non-singular Moreover, x is a critical point of the projection map to the X k coordinate restricted to V, by the inverse function theorem we can choose p coordinates (not including X k ) such that the remaining k p co-ordinates of points of V in a small enough neighborhood U of x are smooth functions of these chosen p co-ordinates Without loss of generality let these p coordinates be X 1,, X p We will denote the remaining co-ordinate functions on U by X p+1 (X 1,, X p ),, X k (X 1,, X p ) noting that they are smooth semi-algebraic functions of X 1,, X p We use that (1) Jac(F, p, 0)(x) has full rank since x is a non-singular point of V, and (2) Hess(X k (X 1,, X p ))(x) is non-singular since x is a non-degenerate critical point with respect to X k Let J = [p + 1, k], and consider the Jacobian matrix Jac(F J, 0, 0) Jac(F J, 0, 0) = F 1 X 1 F 1 X k F k p X 1 F k p X k jac J {1}\{k} X 1 jac J {1}\{k} X k jac J {p}\{k} X 1 jac J {p}\{k} X k Since, by definition of the functions X p+1 (X 1,, X p ),, X k (X 1,, X p ) F i (X 1,, X p, X p+1 (X 1,, X p ),, X k (X 1,, X p )) 0,

12 12 BARONE AND BASU for 1 i k p, by the chain rule for 1 j p, (5) 0 = F 1 X j + F 1 X p+1 X p+1 X j + + F 1 X k X k X j, 0 = F k p X j + F k p X p F k p X k X p+1 X j X k X j Let = det Jac(F, p, p) Notice that in the sub-matrix Jac(F, p, 0) of Jac(F J, 0, 0), for each 1 j p, adding k X i row i (Jac(F, p, 0)) X j i=p+1 to the j-th row and using (5) we can clear out the first p rows Since, rank(jac(f, p, 0)(x)) = k p, this implies that (x) 0 From Cramer s Rule, we have Let for 1 i p, X k = jac J {1}\{k}, X 1 X k = jac J {p}\{k} X p G i (X 1,, X p ) = jac J {i}\{k} (X 1,, X p, X p+1 (X 1,, X p ),, X k (X 1,, X p )) Substituting above we get that From the quotient rule, and in particular X k = G 1(X 1,, X p ), X 1 X k = G p(x 1,, X p ) X p 2 X k X i X j = Hess(X k )(x) = ( 2 X k X i X j (x) G 1 X i G i X j 2, ) 1 i,j p = ( Gi X j (x) (x) ) 1 i,j, p noticing that since x is a critical point of the function X k restricted to V, G 1 (x) = = G p (x) = 0 Applying the chain rule again we have that for 1 i, j p, (6) G i = jac J {i}\{k} jac J {i}\{k} X p+1 jac J {i}\{k} X k X j X j X p+1 X j X k X j

13 ON A REAL ANALOGUE OF BEZOUT INEQUALITY 13 Finally, for each 1 j p, adding k X i row i (Jac(F J, 0, 0)) X j i=p+1 to the j-th row, and using (5) and (6), we see that Jac(F J, 0, 0)(x) is row equivalent to the matrix ( ) 0 Hess(X k ) (x) I k p which is clearly non-singular, since x is a non-degenerate critical point of X k, which implies that the Hess(X k )(x) is non-singular, and we have already observed that (x) 0 Definition 29 Let X P k C be a non-singular variety, and (H µ) µ=(µ0:µ 1) P 1 a C pencil of hyperplanes We call the pencil of varieties (X µ = X H µ ) µ a Lefschetz pencil if it satisfies the two following conditions 1 The base locus B is smooth of co-dimension two in X 2 Each member X µ of the pencil has at most one ordinary double point as a singularity The main result about Lefschetz pencil we will require is the following well known result from complex algebraic geometry (see for example [28, Corollary 210]) Proposition 210 If X P k C is a non-singular variety, then any generic pencil of hyperplane sections of X is Lefschetz Remark 211 Observe that a generic tuple of polynomials G = (G 1,, G k p ) where each G i R[X 1,, X k ] with deg(h i ) = d i and is chosen generically, will have the property that the variety W = Zer(G, P k C ) is non-singular and the pencil of hyperplane sections (W µ = W H µ ) µ indexed by µ = (µ 0 : µ 1 ), where H µ P k C is defined by the equation µ 0 X 0 + µ 1 X k = 0, is a Lefschetz pencil for the variety W by Proposition 210 above Let 0 p k, ε = (ε 1,, ε m ) be a tuple of variables, and P = (P 1,, P k p ), P i R ε [X 1,, X k ] with deg P i d i, and P R[X 1,, X k ], deg P d Let 0 q < p k and G = (G 1,, G k p ) be a tuple of polynomials with G i R[X q+1,, X k ] with deg(g i ) = d i, and G R[X q+1,, X k ] be another polynomial with deg(g) = d, such that 1 The variety W = Zer(G {G}, P k q C ) is a non-singular complete intersection 2 The pencil of hyperplane sections (W µ = W H µ ) µ indexed by µ = (µ 0 : µ 1 ) P 1 C, where H µ P k q C is defined by the equation µ 0 X 0 +µ 1 X k = 0, is a Lefschetz pencil for the variety W Let for every z R q, (7) F z = Def(P, ζ, q, G)(z, ), Def(P, δ, q, G)(z, ) We also need the following notation Notation 212 For 1 p q k, we denote by π [p,q] the projection map on the coordinates X p,, X q, and also denote by R [p,q] the subspace spanned by these coordinates For any set S R k, and z R [1,p] we will denote by S z the fiber S π 1 [1,p] (z)

14 14 BARONE AND BASU Proposition 213 For every z R q, the following holds 1 Def(P, η, q, G)(z, ) h, Def(P, δ, q, G)(z, ) h defines a non-singular complete intersection V z P k q C δ, ε,η of dimension p q 1 2 The pencil of hyperplane sections (V z,µ = V z H µ ) µ indexed by µ = (µ 0 : µ 1 ), where H µ P k q C δ, ε,η is defined by the equation µ 0X 0 + µ 1 X k = 0, is a Lefschetz pencil for the variety V z 3 For each singular point x C k of the pencil (V z,µ ) µ, there exists J [k q+1, k], card J = k p and k J, such that x C J (F z ) (see (4) and (7) for definitions), and x is a simple zero of the system (F z ) J (see (3) for definition) Proof Replacing η and δ by new variables s and t (respectively), and setting s = t = 1 we have that (Def(P, 1, q, G)(z, ) h, Def(P, 1, q, G)(z, ) h ) = (G h, G h ) define a non-singular complete intersection in W P k q C ε (by hypothesis) Moreover, the pencil of hyperplane sections (W µ = W H µ ) µ is Lefschetz by hypothesis Since the property of being a non-singular complete intersection as well as a fixed pencil of hyperplane section being Lefschetz is stable, it also holds for an open neighborhood of the point (s, t) = (1, 1) The set of pairs (s, t) for which any of these two properties is violated is Zariski closed, defined over C ε, and is not the whole of P 1 C ε P1 C ε In particular the complement contains the point (η, δ) (since η, δ are algebraically independent over C ε ) This proves parts 1 and 2 of the proposition Part 3 follows from Proposition 28 We also need the following proposition Proposition 214 Let C be a bounded sa connected component of Bas(P, Q) Then, there exists a subset subset Q Q, and a semi-algebraically connected component D of Zer(P Q, R k ) such that D C Proof See Proposition 131 in [10] Proposition 215 Let F = (F 1,, F k ) be a tuple of polynomials with F i R[X 1,, X k ] and let G = (G 1,, G k ), with G i R[X 1,, X k ], be a tuple of polynomials with deg(g) deg(f) Let x R k a simple zero of F Then, there exists a simple zero x Zer(Def(F, ζ, 0, G), R ζ k ), such that lim ζ x = x Proof It follows from the fact that x is a simple zero of the family F that any infinitesimal perturbation of the family F will have a simple zero, x C ζ k, in an infinitesimal neighborhood of x To see this observe that since x R k (and is thus in particular bounded over R), it belongs to the image under the lim ζ map (extended to elements of C ζ k which are bounded over R) of Zer(Def(F, ζ, 0, G), C ζ k ) Thus, there exists x Zer(Def(F, ζ, 0, G), C ζ k ), such that lim ζ x = x Moreover, since x is a simple zero of Zer(F, R k ), we have that det(jac(f, 0, 0))(x) 0 (see Notation 27) Since, lim ζ this implies that (det(jac(def(f, ζ, 0, G), 0, 0))( x)) = det(jac(f, 0, 0))(x), det(jac((def(f, ζ, 0, G), 0, 0)))( x) 0 as well, since det(jac(f, 0, 0))(x) R \ {0}, and hence x is a simple zero of Def(F, ζ, 0, G)

15 ON A REAL ANALOGUE OF BEZOUT INEQUALITY 15 Moreover, x must belong to R ζ k as long as the perturbed polynomials also have real coefficients Otherwise, since complex zeros must occur in conjugate pairs, if x R ζ k, then x x, while lim ζ x = lim ζ x = x, and this implies that x is not a simple zero of F 22 Generic coordinates We recall in this section a result proved in [4] that we will require Notation 216 For a real algebraic set V = Zer(Q, R k )we let reg(v ) denote the non-singular points in dimension dim V of V (Definition 339 in [12]) Definition 217 Let V = Zer(Q, R k ) be a real algebraic set Define V k = V, and for 0 i k 1 define Let d V (i) denote the dimension of V (i) V (i) = V (i+1) \ reg(v (i+1) ) Definition 218 Let V = Zer(Q, R k ) be a real algebraic set, 1 j k, and l Gr(k, k j) We say that the linear space l is j-good with respect to V if either: j d V ([0, k]), or d V (i) = j, and the set {x reg(v (i) ) dimt x V (i) l = 0} is a non-empty dense Zariski open subset of V (i) Definition 219 Let V = Zer(Q, R k ) and B = {v 1,, v k } be a basis of R k We say that the basis B is generic with respect to V if for each j, 1 j k, the linear space span(v 1,, v k j ) is j-good with respect to V The following proposition appears in [4] Proposition 220 Let V = Zer(Q, R k ) and {v 1,, v k } be a basis of R k Then, there exists a non-empty open semi-algebraic subset U of linear transformations GL(k, R) such that for every T U the basis {T (v 1 ),, T (v k )} is generic with respect to V 3 Proofs of the main theorems We now fix polynomials Q 1,, Q l and and the varieties V 1,, V l as in Theorem 4 We will assume if necessary by initially squaring each polynomial that each Q i is non-negative over R k Since this increases each degree by a multiplicative factor of 2, this does not affect the asymptotics of the bound The section is organized as follows In Subsection 31 we define certain approximating semi-algebraic sets and prove their important properties In Subsection 32 we recall and then apply in the current context certain well-known bounds on the Betti numbers of non-singular cmplete intersections Finally, we prove the main theorems of this paper in Subsections 33 and 34

16 16 BARONE AND BASU 31 Definitions and main properties of approximating semi-algebraic sets We first introduce in 311 some necessary notation, and then in Subsection 312 below we describe the construction of certain semi-algebraic sets approximating the varieties V j The main properties of these sets is then proved in Subsection 313 The approximating properties of these sets are proved in Proposition 313, and the quantitative estimates on the degrees of the polynomials appearing in the description of these approximating sets is proved in Proposition Notation Notation 31 For any semi-algebraic set S and x S, we denote by dim x S the local dimension of S at x For 0 j l and x V j, we denote dim (j) (x) = (dim x V 1,, dim x V j ) Notation 32 We will use the natural partial order on the sets N j, and denote for σ = (σ 1,, σ j ), τ = (τ 1,, τ j ) N j, σ τ, if σ i τ i for all 1 i j Before proceeding further we illustrate the notation introduced above by considering some examples We first consider again Example 18 from Section 1 In this example, (following Notation 19) we have V 1 = Zer(X 3, R k ), V 2 = Zer(X 3, R k ), V 3 = {1,, d} 3 The various functions dim (j) : V j N j, j = 1, 2, 3 (cf Notation 31) are as follows dim (1) (x) = (2) if x V 1, dim (2) (x) = (2, 2) if x V 2 = V 1, dim (3) (x) = (2, 0) if x V 3 The next example is slightly more involved but is helpful in understanding the proof of Proposition 313 below Example 33 Let k = 4, l = 3, and Q 1 = (X 1 + X 2 + X 3 + X 4 )(X X 2 4 ), Q 2 = X X 2 4, Q 3 = X X (X 2 1 X 3 2 ) 2 We denote by (e i ) 1 i 4 the elementary basis vectors in R 4 Denote by L 1 the hyperplane defined by X 1 + X 2 + X 3 + X 4 = 0, by L 2 = span(e 1, e 2 ), the linear subspace defined by X 3 = X 4 = 0, and by C 3 the cubic curve contained in L 2, defined by the equation X 2 1 X 3 2 = 0 Notice that L 1 L 2 is the line in span(e 1, e 2 ) defined by X 1 + X 2 = 0, and it meets C 3 at the points x 0 = 0 (which is a singular point of C 3 ), and x 1 = ( 1, 1, 0, 0, 0) (which is a regular point of C 3 ) These sets are depicted in Figure 1

17 ON A REAL ANALOGUE OF BEZOUT INEQUALITY 17 Figure 1 The sets L 1, L 2 and C 3 in Example 33 restricted to the hyperplane X 4 = 0 We have using Notation 19, V 1 = L 1 L 2, V 2 = L 2, V 3 = C 3 The various functions dim (j) : V j N j, j = 1, 2, 3 can now be described as follows dim (1) (x) = (3) if x L 1, dim (1) (x) = (2) if x L 2 \ L 1, dim (2) (x) = (3, 2) if x L 2 L 1, dim (2) (x) = (2, 2) if x L 2 \ L 1, dim (3) (x) = (3, 2, 1) if x = x 0, x 1, dim (3) (x) = (2, 2, 1) if x C 3 \ {x 0, x 1 } Remark 34 Observe that in Example 33 above, the point x 0 = 0 is a singular point of V 3, and x 0 V (3,2,1) However, for any open neighborhood U V 3 of x 0 in V 3, and x U \ {x 0 }, we have that x is a regular point of V 3, and if moreover x x 1, dim (3) (x ) = (2, 2, 1) < (3, 2, 1) (cf Notation 32) Notice also that (2, 2, 1) is a minimal element of the set {dim (3) (x) x V 3 }, and the semi-algebraic subset of V 3 defined by {x V 3 dim (3) (x) = (2, 2, 1)} = V 3 \ {x 0, x 1 } is open in V 3 (cf Proposition 35 below) The following property of the function dim (j) : V j N j will be important later Following Notation 19 as before we have the following proposition

18 18 BARONE AND BASU Proposition 35 Let σ N j, 1 j l Then, the semi-algebraic subset V σ j V j defined by V σ j = {x V j dim (j) (x) σ} is open in V j In particular, if U V j is an open semi-algebraic subset of V j, and σ N j is such that σ is a minimal element of the set {dim (j) (x) x U}, then the semi-algebraic subset {x U dim (j) (x) = σ} is open in V j Proof The proof is by induction on j If j = 1, then the proposition follows immediately from the upper semi-continuity property of the dimension function Now suppose that the proposition is true for all smaller values of j Let σ = (σ 1,, σ j ) and let σ = (σ 1,, σ j 1 ) Using the induction hypothesis we have that V σ j 1 is open in V j 1 This implies that there exists an open semi-algebraic subset U R k, such that V σ j 1 = V j 1 U Also, the semi-algebraic set V σj j = {x V j dim x (V j ) σ j } V j is open in V j Thus, there exists an open semi-algebraic set U R k, such that V σj = V j U Now, V σ j = V σj j V σ j 1 = (V j U ) (V j 1 U) = V j U U (since V j V j 1 ) Hence, V σ j is open in V j Notation 36 For 0 j l we call τ = (τ 1,, τ j ) N j admissible if it satisfies the following two conditions 1 τ 1 τ j, 2 for 1 i < j, τ i k i We denote the subset of admissible tuples of N j by A j, and denote by A the set A l For σ = (σ 1,, σ j ), τ = (τ 1,, τ j ) A j, we say σ τ, if σ i τ i for each i, 1 i j Notation 37 For each j, 1 j l, we denote by R j the real closed field R δ j,, δ 1, η 1, ζ 1,, η j, ζ j Notice that R j is a real closed extension of the field R j 1 For any semi-algebraic subset S R j, we will denote by S b the union of semi-algebraically connected components of S which are bounded over R Remark 38 For readers familiar with arguments in real algebraic geometry involving multiple infinitesimals, this ordering of the infinitesimals in Notation 37 might seem somewhat counter-intuitive, since we will consider the varieties V i s in the order V 1, V 2, etc, and the infinitesimal δ i will be used to perturb the variety V i, one would expect that the infinitesimals δ i s to be ordered the other way round The reason behind this ordering of the infinitesimals will become clear in the proof of Proposition 313 below 312 Definition of sequences of approximating semi-algebraic sets We now describe the construction of certain semi-algebraic sets approximating the varieties V j We assume that V 1, and hence each V j, are bounded over R We will also use the following notation

19 ON A REAL ANALOGUE OF BEZOUT INEQUALITY 19 Notation 39 For any set X and j 0 we will denote by ( X j ) the set of all subsets of X of cardinality j Definition 310 For any τ A j we define an index set I j (τ), and a family (Vτ,j α Rk j ) α I j(τ) as follows Each Vτ,j α = ( Bas(Pτ,j α, {Qα τ,j })), where b Pα τ,j, is an ordered tuple of polynomials, and Q α τ,j R j[x 1,, X k ] defined inductively as follows 1 If j = 0, then for τ = (), define I 0 (τ) = { 1}, and P ( 1) τ,0 = (0), Q ( 1) τ,0 = 0 2 Otherwise, we denote by τ = (τ 1,, τ j 1 ) and let p = τ j 1, q = τ j Let G be a generic polynomial in R[X q+1,, X k ] strictly positive over R k q with deg(g) = deg( P j ), P j = Q i R[X 1,, X k ], 3 1 i j P j = Def( P j, δ j, q, H) R δ j [X 1,, X k ] I j (τ) = I j 1 (τ ) { 1}, if τ j 1 = τ j, ( ) = I j 1 (τ [τj + 1, k] ), else k τ j (where denotes the usual Cartesian product) 4 For each triple (α I j 1 (τ ), P α τ,j 1, Qα τ,j 1 ) if τ j 1 = τ j, then denoting β = (α, 1) let (8) (9) otherwise, suppose that P β τ,j = P α τ,j 1, Q β τ,j = P j P = P α τ,j 1 = (P 1,, P k p ) R j 1 [X 1,, X k ] k p, with deg(p i ) = d i, for 1 i k p, and d = (d 1,, d k p ) Let G = (G 1,, G k p ) be generic polynomials in R[X q+1,, X k ] with deg(g i ) = d i and strictly positive over R k q, 1 i k p We define (using Notation 26) P = Def(P, η j, q, G) F = ( P, P j ) Finally, for each J ( [τ j+1,k] k τ j 1+1), denoting β = (α, J), and following the notation introduced above (and using Notation 27) P β τ,j = Def(F J, ζ j, k, G ), Q β τ,j = Q α τ,j 1, where G = (G 1,, G k q ) is another tuple of generic polynomials strictly positive over R k with deg(g ) = ( d α, d j, d,, d ), where d = (k p + 1)d j and d α = deg(p)

20 20 BARONE AND BASU Notation 311 For each j, 0 j l, τ A j, let (cf Notation 31) Ṽ τ = Using Notation 311: α I j(τ) V α τ,j, V τ = {x V j dim (j) (x) = τ} Proposition 312 For each j, 1 j l, V j = V τ τ=(τ 1,,τ j) A j,τ j k j Proof This is immediate from the definition of A j and the various V τ, τ = (τ 1,, τ j ) A j, τ j k j, and the fact that dim V i k i for 0 i j 313 Properties of the approximating sets The following proposition and its corollary guarantees the approximating properties of the sets Vτ,j α defined above and is the main technical proposition of the paper Assume that the given system of coordinates is generic with respect to the finite number of varieties V τ (cf Proposition 220) Proposition 313 For all τ = (τ 1,, τ j ) A j, with τ j k j, V τ W τ V j, where W τ = σ lim δj Ṽ σ, and the union is taken over all σ A j with σ j = τ j, and σ i τ i for all 1 i < j In the proof of Proposition 313 we need the following technical lemma that we prove first We draw the attention of the reader to the ordering of the infinitesimals in this lemma, which is particularly delicate and plays a very important role in the proof of the lemma In particular, notice that if ε = (ε 1,, ε m ), δ are variables, then we have the following diagram of real closed subfields of the real closed field R δ, ε R δ, ε R δ R ε R In particular, if V (respectively, Z) is a semi-algebraic subset of R ε k (respectively, R δ k ), then Ext(V, R δ, ε ), Ext(Z, R δ, ε ) (recall Notation 22) are semialgebraic subsets of R δ, ε k

21 ON A REAL ANALOGUE OF BEZOUT INEQUALITY 21 Figure 2 Example illustrating Lemma 314 Lemma 314 Let P, H R[X 1,, X k ], P non-negative, and H strictly positive at all points of R k Let V R ε k be a semi-algebraic set bounded over R, where ε = (ε 1,, ε m ) Let P = (1 δ)p δh, and Z a semi-algebraically connected component of Zer( P, R δ k ), such that Z = Zer( P, R δ k ) B k (x, r), for some x R k and r > 0, r R Suppose that lim ε1 (Ext(V, R δ, ε )) Z Then, Ext(V, R δ, ε ) Ext(Z, R δ, ε ) Before proving Lemma 314 we illustrate it with a simple example Example 315 In this example, k = 2, and V = Zer((X X 2 2 1) 2 ε, R ε 2 ) (shown in blue in Figure 2), P = (X 1 1) 2 + X 2 2, H = 1 We display the various sets occurring in this example in Figure 2 after choosing ε, δ, to be certain sufficiently small positive real numbers, with ε δ It is clear from definition that P is non-negative, Zer(P, R k ) consists of the single point (1, 0) (shown in black in Figure 2), and the variety Zer( P, R δ 2 ), where P = (1 δ)p δh, has one semi-algebraically connected component, Z, which is also depicted in black The semi-algebraic set lim ε V = Zer((X X 2 2 1) 2, R 2 ) is the unit circle centered at the origin (shown in red), and Ext(V, R δ, ε ) meets Ext(Z, R δ, ε ) in two points, and Ext(V, R δ, ε ) Ext(Z, R δ, ε ) is not empty, and consists of four points as can be seen in Figure 2 P P +H Proof of Lemma 314 Let G R(X 1,, X k ) denote the rational function which is continuous, and takes non-negative values at all points of R k by hypothesis Let y Ext(V, R δ, ε ) be such that z = lim ε1 y Z Since, Z is contained in B k (x, r), and y Ext(V, R δ, ε ) is ε 1 -infinitesimally close to z Z, it is clear that Ext(V, R δ, ε ) B k (x, r) contains y and in particular is not empty Let C be the semi-algebraically connected component of Ext(V, R δ, ε ) B k (x, r) which contains y

22 22 BARONE AND BASU Figure 3 The variety V 1 We prove that Ext(C, R δ, ε ) Ext(Z, R δ, ε ) Suppose otherwise Then, G(y) δ Suppose without loss of generality that G(y) δ > 0 Since, z = lim ε1 y Zer( P, R δ k ), it is clear that lim ε1 (G(y) δ) = 0 Let h = inf x C G(x) Since, C is a semi-algebraic set defined over R ε, and G is a continuous rational function defined over R, it follows that h R ε Moreover, since Ext(C, R δ, ε ) Zer( P, R δ, ε k ) =, G(y) δ > 0, and C is closed and bounded, the infimum of G over C is achieved at a point, and hence h > δ On the other hand, from the fact that lim ε1 (G(y) δ) = 0, it follows that lim ε1 h = δ This is impossible, since lim ε1 h R Lemma 316 Suppose that σ = (σ 1,, σ j ), τ = (τ 1,, τ j ) A j with σ i τ i, 1 i < j, and σ j = τ j Then, W σ W τ Proof Obvious from the definitions of W σ and W τ Before giving the proof of Proposition 313 we consider the following threedimensional example which illustrates some of the finer points Example 317 Let k = 3, l = 3, and Q 1 = (X X X 2 3 1)(X (X X2 2 1) 2 ), Q 2 = (X (X X2 2 1) 2 ), Q 3 = X X (X 1 1) 2 The variety V 1 (shown in Figure 3) is bounded, and equal to the union of the unit sphere S R 3 (shown in ornage), and an ellipse, C (shown in green), contained in the plane span(e 1, e 2 ), with S C = {(±1, 0, 0)} (shown in red) The variety V 2 = C, and V 3 = {(1, 0, 0)} The various functions dim (j) : V j N j are as follows