A Geodesic Stratification of Two-dimensional Semi-algebraic Sets
AA GEODESIC STRATIFICATION OF TWO-DIMENSIONALSEMI-ALGEBRAIC SETS
CHENGCHENG YANG
This paper is dedicated to my parents and my thesis advisor Dr. Robert Hardt. My parents have supported meunconditionally. My advisor has given me many valuable advices. A BSTRACT . Given any arbitrary semi-algebraic set X , any two points in X may be joinedby a piecewise C path γ of shortest length. Suppose A is a semi-algebraic stratificationof X such that each component of γ ∩ A is either a singleton or a real analytic geodesicsegment in A , the question is whether γ ∩ A has at most finitely many such components.This paper gives a semi-algebraic stratification, in particular a cell decomposition, of areal semi-algebraic set in the plane whose open cells have this finiteness property. Thisprovides insights for high dimensional stratifications of semi-algebraic sets in connectionwith geodesics. Date : February 19, 2021.2020
Mathematics Subject Classification.
Key words and phrases. geodesics, real algebraic sets, real semi-algebraic sets, cell decomposition.The author was support in part by NSF Award a r X i v : . [ m a t h . AG ] F e b CHENGCHENG YANG INTRODUCTION
A semi-algebraic set X in the plane can be described as: X = I (cid:91) i = J (cid:92) j = { ( x , y ) ∈ R : f i , j ( x , y ) = , g i , j ( x , y ) > } , where f i , j , g i , j are polynomials in two variables. We see that X is a finite union of sets in theform obtained by taking the intersection of an algebraic set (i.e. { f ( x , y ) = , . . . , f k ( x , y ) = } ) with an open semi-algebraic set (i.e. { g ( x , y ) > , . . . , g m ( x , y ) > } ).The triangulability question for algebraic sets was first considered by van de Waerdenin 1929 [8]. It is a well-known theorem that every algebraic set is triangulizable [2]. Onthe other hand, in 1957 Whitney introduced another splitting process that divides a realalgebraic V into a finite union of “partial algebraic manifolds” [9]. An algebraic partialmanifold M is a point set, associated with a number ρ , with the following property. Takeany p ∈ M . Then there exists a set of polynomials f , . . . , f ρ , of rank ρ at p , and a neigh-borhood U of p , such that M ∩ U is the set of zeros in U of these f i . The splitting processuses the rank of a set S of functions f , . . . , f s at a point p , where the rank of S at p is thenumber of linearly independent differentials d f ( p ) , . . . , d f s ( p ) .In 1975 Hironaka reproved that every semi-algebraic set is also triangulable and alsogeneralized it to sub-analytic sets [2]. His proof came from a paper of Lojasiewicz in1964, in which Lojasiewicz proved that a semi-analytic set admits a semi-analytic trian-gulation [4]. In 1975, Hardt also proved the triangulation result for sub-analytic sets byinventing another new method [3]. Since any semi-algebraic set is also semi-analytic, thusis sub-analytic, both Hironaka and Hardt’s results showed that a semi-algebraic set admitsa triangulation, that is to say, it is homeomorphic to the polyhedron of some simplicialcomplex.Following the examples of Whitney’s stratification and Lojasiewicz/Hironaka/Hardt’striangulation, this paper tries to build a cell-complex stratification such that it admits ananalytical condition concerning shortest-length curves. The idea is explained more pre-cisely as follows.Suppose A , B are two arbitrary points in X , and γ is a piecewise C curve from A to B lying entirely in X such that its length is the shortest among all possible such curves. Wesearch for a semi-algebraic cell decomposition (that is each cell is a semi-algebraic set in R ) A of X , such that the intersection of γ with every cell in A is either empty or consistsof finitely many components, each of which is either a singleton or a geodesic line segment.We will explain the meaning of a geodesic line segment soon. If a cell decomposition A satisfies this property, we will simply say that A satisfies the finiteness property .Our first step is to assume that X is an arbitrary affine algebraic variety, that is,(1.1) X = { f ( x , y ) = , . . . , f k ( x , y ) = } , k ≥ , where the f i ( x , y ) are nonzero distinct polynomials. We argue that a cell decomposition A exists with the finiteness property, and ( X , A ) can be shown to be a CW complex. Moregenerally, we may assume that X is a finite union of sets in the above form, then the sameconclusion holds.Our next step is to look at an open planar semi-algebraic set X in the form of:(1.2) X = { g ( x , y ) > , . . . , g m ( x , y ) > } , m ≥ , GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 3 where the g j ( x , y ) are nonzero distinct polynomials. More generally, we may assume that X is a finite union of sets in the above form. A cell decomposition for such an X can alsobe established with the desired finiteness property which is also a CW decomposition.Our third step is to search for a CW decomposition with the desired finiteness propertyfor X , which is a finite union of sets in the form of:(1.3) { f ( x , y ) = , . . . , f k ( x , y ) = , g ( x , y ) > , . . . , g m ( x , y ) > } , k ≥ , m ≥ . Our final step is to take a finite union of sets in the previous steps: (1.1) and (1.2); (1.1)and (1.3); (1.2) and (1.3); (1.1) and (1.2) and (1.3).
CHENGCHENG YANG
2. T
HE STRATIFICATION OF AN AFFINE ALGEBRAIC SET IN R Suppose X = { ( x , y ) ∈ R : f ( x , y ) = } is an affine algebraic set in the plane. Suppose p = ( x , y ) is a nonsingular point of X , that is d f ( p ) (cid:54) =
0. Without loss of generality as-suming that ∂ f ∂ y ( p ) (cid:54) =
0, the implicit function theorem implies that there exist open intervals I , J of x , y , respectively, and a differentiable function g : I → J such that g ( x ) = y and { ( x , y ) ∈ I × J | f ( x , y ) = } = { ( x , g ( x )) ∈ R | x ∈ I } [7]. So we obtain a smooth parametrization g for X in an open neighborhood of p . In thepaper [10], we’ve shown how to construct a cell decomposition with the desired finitenessproperty in the closed region below the graph of g under the assumption that g is a poly-nomial function. More generally, the closed region could be replaced by an open regionbelow the graph, and the polynomial function could be replaced by a smooth function withfinitely many strict inflection and local minimum points. The following lemma verifies thatg is in fact a real analytic (thus smooth) function over the open interval I . Lemma 2.1.
Suppose g is the differentiable function given as before by the implicit func-tion theorem for the polynomial function f ( x , y ) = at p = ( x , y ) , where ∂ f ∂ y ( p ) (cid:54) = , theng is a real analytic function over the open interval I.Proof. Since f ( x , y ) is a polynomial in two variables, we can consider the complex polyno-mial function f ( z , w ) , where z , w are variables in C . It follows that f ( z , w ) is a holomorphicfunction in two variables. Now we can apply the holomorphic implicit function theorem,since ∂ f ∂ w (cid:54) = ( x , y ) ∈ C × C , the equation f ( z , w ) = w ( z ) in a neighborhood x that satisfies w ( x ) = y [1]. Hence g ( x ) = w ( x ) for x ∈ I when I is in this neighborhood. (cid:3) Corollary 2.2. If [ a , b ] is a closed and bounded interval contained in I and suppose g isnot a linear function, then g has finitely many inflection and local minimum points over [ a , b ] .Proof. For local minimum points (more generally, critical points), differentiating bothsides of the equation f ( x , g ( x )) = f x ( x , g ( x )) + f y ( x , g ( x )) · g (cid:48) ( x ) = , where we use f x , f y as short-hand notations for ∂ f ∂ x , ∂ f ∂ y , respectively. It implies that g (cid:48) ( x ) = f x ( x , g ( x )) =
0, because we may assume that f y ( x , g ( x )) (cid:54) = x ∈ I bycontinuity.Since f x is also a polynomial, f x ( x , g ( x )) is a real analytic function over the interval I .Therefore f x ( x , g ( x )) has isolated zeros unless it is identically equal to zero in which case g is a constant function. Since [ a , b ] is a compact interval, there are at most finitely manyzeros of f x ( x , g ( x )) over [ a , b ] , thus there are at most finitely many local minimum points(or critical points) of g over [ a , b ] .Similarly, for inflection points, we differentiate equation (2.1) again to obtain the fol-lowing equation:(2.2) f xx ( x , g ( x )) + f xy ( x , g ( x )) · g (cid:48) ( x ) + f yy ( x , g ( x )) · g (cid:48) ( x ) + f y ( x , g ( x )) · g (cid:48)(cid:48) ( x ) = . It follows that g (cid:48)(cid:48) ( x ) = f xx ( x , g ( x )) + f xy ( x , g ( x )) · g (cid:48) ( x ) + f yy ( x , g ( x )) · g (cid:48) ( x ) = , which is real analytic and so has at most finitely many zeros over any compact interval [ a , b ] because g (cid:48)(cid:48) is not identically zero by hypothesis. (cid:3) GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 5
Now we are ready to give a cell decomposition for a compact and connected algebraicvariety of an irreducible polynomial in two real variables.
Theorem 2.3.
Suppose f ( x , y ) is an irreducible polynomial function and X = V ( f ) is theaffine algebraic variety determined by the zeros of f . Assume X is compact and connected,then X has a cell decomposition A with the desired finiteness property. The proof immediately gives the following corollary.
Corollary 2.4.
Given the cell decomposition A as in the theorem, if γ is any shortest-length piecewise C -curve between two points in X, then the intersection of γ with any0-cell in A is either empty or a singleton; the intersection of γ with any 1-cell in A iseither empty, or a continuous line segment contained in the 1-cell.Remark . A continuous line segment contained in a 1-cell is one example of a geodesicline segment. In general, a line segment is said to be geodesic if it is either a straight linesegment or a continuous line segment contained (partially or entirely) inside a 1-cell.
Proof. If f is a polynomial in x - (or y -)variable only, then X has at most one zero and thetheorem follows trivially. Without loss of generality, we may assume that f is a polynomialfunction that has both x and y variables. In particular, f x and f y are not zero. Then wehave that f and f x share no common factors, because f is irreducible by hypothesis. Analgebraic geometry theorem says if k is an arbitrary commutative field, and F , G ∈ k [ x , y ] arenonzero polynomials without common factors, then V ( F ) ∩ V ( G ) is finite [6]. Applyingthis theorem, we conclude that(2.3) V ( f ) ∩ V ( f x ) is finite.Similarly, the same reasoning implies that(2.4) V ( f ) ∩ V ( f y ) is finite.In particular, the set S of singular points in X , that is S = { x ∈ X : where ∂ f ∂ x ( x ) = ∂ f ∂ y ( x ) = } consists of at most finitely many points. Case 1: suppose S is an empty set, then at every p = ( x , y ) ∈ X , either f x (cid:54) = f y (cid:54) = p has an open neighborhood I × J whose intersectionwith X is the graph of a real analytic function over either I or J . Shrinking I and J ifnecessary, we may also assume that the intersection of the closed neighborhood ¯ I × ¯ J with X is the graph of a real analytic function.Since X is compact, X can be covered by finitely many such open neighborhoods, say I × J , . . . , I r × J r , where I i × J i (cid:54)⊂ I j × J j for i (cid:54) = j . In each intersection of X with ¯ I i × ¯ J i ,the graph has finitely many (strict) inflection and local minimum points, thus giving a celldecomposition as we’ve shown in [put a book citation here]. More precisely in this specialcase, the 0-cells are the (strict) inflection and local minimum points, together with the twoendpoints; and the 1-cells are the graphs in between them. Therefore, we find a finite celldecomposition for X with the desired finiteness property. Case 2: suppose S is not an empty set, we know that S consists of finitely many points,thus each of which is an isolated point in X . Let’s pick an open ball B ( q ) for each q ∈ S such that B ( q ) contains no other point in S . Furthermore, in virtue of (2.3) and (2.4) and CHENGCHENG YANG shrinking B ( q ) if necessary, we may assume that for every point p in the closed ball ¯ B ( q ) , p (cid:54) = q , we have(2.5) ∂ f ∂ x ( p ) (cid:54) = , and ∂ f ∂ x ( p ) (cid:54) = . Again by the compactness of X , X can be covered by finitely many open sets in the formof either I × J around a non-singular point p or B ( q ) for a singular point q in S . For theintersection of X with ¯ I × ¯ J , we use the same cell decomposition as shown in case 1 above.For the intersection of X with B ( q ) , we need to first prove the following lemma. Lemma 2.6.
Under the same assumption as before, the intersection of the punctured ballB ( q ) \ { q } , for each q ∈ S, with X consists of finitely many connected components, each ofwhich is homeomorphic to the real line R .Proof. Since B ( q ) \ { q } is an open subset of R , its intersection with X is an open subsetof X , thus consisting of open connected components. Each connected component is locallyEuclidean due to (2.5). Using the subspace topology inherited from R , each connectedcomponent is also Hausdorff and second-countable. Therefore, each connected componentis a connected 1-dimensional manifold. By the classification theorem for connected 1-manifolds, each connected component is homeomorphic to S if it is compact, and R ifit is not [5]. Suppose there exists a connected component P that is homeomorphic to S ,then P is compact in X , thus closed in X (by the Hausdorff property of X ). Because P is also open in X , it follows that P is both open and closed in X . By hypothesis X isconnected, so X = P . However, q is also in X and q is not contained in P , so X (cid:54) = P ,which is a contradiction. So every connected component in the intersection of B ( q ) \ { q } is homeomorphic to the real line R .Next we want to show that there are finitely many such components. Given a connectedcomponent P , P is contained in the punctured open ball B ( q ) \ { q } , so its boundary iscontained in the boundary of B ( q ) \ { q } , which is ∂ B ( q ) ∪ { q } . There are four possibilitiesfor the two endpoints of P : they are on the circle ∂ B ( q ) and are the same; they are on thecircle but not the same; one of them is on the circle and the other is q ; they are both equalto q (see Figure 1). qB(q) qB(q) qB(q) qB(q)(i) (ii) (iii) (iv) F IGURE
1. Four possibilities of a connected component in B ( q ) \ { q } ∩ X .(i) Let’s start with showing that the first situation is impossible. If the two endpoints of P are both equal to a point p on the boundary of B ( q ) , according to ( 2.5), the closure ¯ P of P is locally Euclidean, thus is homeomorphic to S , because it is also closed and bounded.¯ P is closed in X . Furthermore, there exists an open neighborhood I × J around p such that GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 7 the intersection of X with I × J is equal to ¯ P ∩ X . It follows that ¯ P is also an open subsetof X . Therefore, X = ¯ P , since X is connected by hypothesis, which yields a contradictionsince q / ∈ ¯ P .(ii) We show that there are at most finitely many components whose two endpoints areon the boundary of B ( q ) which are not the same. Suppose for the sake of contradictionthat there are infinitely many such components, then their endpoints are infinitely many,because each point on ∂ B ( q ) is an endpoint of at most two connected components. Bysequential compactness of ∂ B ( q ) , there exists a subsequence { p j } ∞ j = of these endpointsthat converges to a point p in ∂ B ( q ) , which is also in X for X is closed. Since p is non-singular, there exists an open neighborhood I × J of p such that the intersection of I × J with X is the graph of a real analytic function g . Then g intersects the circle ∂ B ( q ) infinitelymany times near p . Because ( − t ) / has a power series expansion for | t | <
1, and itconverges absolutely and uniformly on compact subsets of ( − , ) , thus ( − t ) / is realanalytic over the open interval ( − , ) ([put a book citation here, Folland, exercise 66, p.139]). It follows that the arc near p can be parametrized by a real analytic function aswell. Since their difference is also real analytic and they have a zero that is not isolated,the graph of g coincides with the circle near p . This implies that near p there cannot beany endpoint of a connected component, thus leading to a contradiction.(iii) Similarly, there are at most finitely many components whose endpoints are made ofone point on the boundary of B ( q ) and one point being q .(iv) We finish the proof of the lemma by showing that the case when the two endpointsof P are both equal to q does not happen as well. Since ∂ f ∂ x and ∂ f ∂ y are both nonzero forevery point in B ( q ) \ { q } , each point in P has an open neighborhood I × J such that theintersection of X with I × J is not only the graph of a real analytic function g ( x ) for x ∈ I ,but also the graph of a real analytic function h ( y ) for y ∈ J . It follows that for each x ∈ I , x satisfies the equation x = h ( g ( x )) , thus implying(2.6) 1 = h (cid:48) ( y ) · g (cid:48) ( x ) , for y = h ( x ) , and x ∈ I . So g (cid:48) ( x ) is either > < I . Without loss of generality, let usassume that g (cid:48) > P . Then consider the set S of all points in P satisfying the same property. That is, S = (cid:8) r ∈ P : r has an open neighborhood I × J such that X ∩ I × J is the graph ofsome real analytic function g ( x ) , and g (cid:48) ( x ) > x ∈ I (cid:9) . P being path-connected implies that S = P , because it is easy to see that S is both openand closed, and S is also non-empty. Since P is homeomorphic to R , we can choose anorientation for P where locally the graph is increasing as we move along this direction.Thus, starting from q and following this orientation, the y-coordinate always increases,therefore it is impossible that the other endpoint of P returns to q .Finally, since only cases (ii) and (iii) are allowed, and there can be at most finitely manyconnected components in each case, we prove the lemma. (cid:3) Let’s continue proving the theorem. According to Lemma 2.6, since there are at mostfinitely many connected components in the intersection of X with B ( q ) \ { q } , we mayshrink the open ball B ( q ) if necessary to make sure that no component in case (ii) appears.Therefore, it remains to describe a cell decomposition for each component P in case (iii)of the lemma.Before proceeding with the description, we need to demonstrate the following lemma. CHENGCHENG YANG
Lemma 2.7.
Suppose P is a connected component in the intersection of X with B ( q ) \ { q } ,and one endpoint of P is equal to q, and the other is on the boundary of B ( q ) , then P isthe graph of a real analytic function g ( x ) for x over an open interval I = ( a , b ) , where − ∞ < a < b < ∞ . Moreover, g ( x ) has no critical points and at most finitely many strictinflection points over the interval I.Proof. For every point r on P , r has an open neighborhood I r × J r such that the intersectionof X with I r × J r is the graph of a real analytic function g r ( x ) for x ∈ I r . Moreover, we mayassume that I r × J r is contained inside the open ball B ( q ) . Let I = (cid:91) r ∈ P I r . Then we show that I is connected, thus I is an open interval. Suppose not, there exist twodisjoint subsets P , P of P , such that P ∪ P = P , and I = (cid:0) (cid:91) r ∈ P I r (cid:1) (cid:91) (cid:0) (cid:91) r ∈ P I r (cid:1) , where the union is disjoint. We can deduce that (cid:91) r ∈ P I r × J r and (cid:91) r ∈ P I r × J r are disjoint.Since P = P ∩ (cid:91) r ∈ P I r × J r = (cid:0) P ∩ (cid:91) r ∈ P I r × J r (cid:1) (cid:91) (cid:0) P ∩ (cid:91) r ∈ P I r × J r (cid:1) , P turns out to be a disconnected set, which is a contradiction to the hypothesis that P is aconnected component. Thus I is connected. Since I is also open and bounded (under theextra assumption that each I r × J r is contained in B ( q ) ), there exist two real numbers a < b such that I = ( a , b ) .Given two distinct points r , r in P , suppose that I r and I r overlap nontrivially, thenwe claim that g r and g r agree over the intersection I r ∩ I r . The proof of the claim isessentially the same as shown in (2.6), except that in this case we look at the x -coordinateinstead of the y -coordinate. As a result, P can have one and only one graph over each pointin the common interval of I r and I r . It follows that g r = g r there.Given this, for each x ∈ I , if we define the value of g ( x ) to be g r ( x ) whenever x ∈ I r forsome r ∈ P . Then g is well-defined over the entire open interval I . Furthermore, g is a realanalytic function.(The following was first suggested to me by Dr. Hardt, which significantly simplifiesthe cell decomposition. My original cell decomposition involves infinitely many cells neara singular point.)Since g (cid:48) ( x ) is either > < I , g has no critical point over I .For the inflection points, since ∂ f ∂ x and ∂ f ∂ y are both nonzero in B ( q ) \ { q } , equation (2.1)implies that g (cid:48) ( x ) = − f x ( x , g ( x )) f y ( x , g ( x )) . Substituting this into equation (2.2), we obtain an expression for g (cid:48)(cid:48) ( x ) : g (cid:48)(cid:48) ( x ) = f x ( x , g ( x )) f y ( x , g ( x )) f xy ( x , g ( x )) − f y ( x , g ( x )) f xx ( x , g ( x )) − f x ( x , g ( x )) f yy ( x , g ( x )) f y ( x , g ( x )) . It follows that if g (cid:48)(cid:48) ( x ) = ( x , g ( x )) is contained in the following variety: V ( f ) ∩ V ( f x f y f xy − f y f xx − f x f yy ) . GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 9
In the case that the polynomial 2 f x f y f xy − f y f xx − f x f yy is zero or is divisible by f , g (cid:48)(cid:48) is identically equal to zero thus having no strict inflection point. Otherwise, under the as-sumption that f is irreducible, f and 2 f x f y f xy − f y f xx − f x f yy are two nonzero polynomialsin R [ x , y ] with no common factors, so the variety V ( f ) ∩ V ( f x f y f xy − f y f xx − f x f yy ) is afinite set. Therefore, g (cid:48)(cid:48) ( x ) has at most finitely many strict inflection points over the openinterval I . (cid:3) We finish the proof Theorem 2.3 as follows. According to Lemma 2.7, a component P in case (iii) is the graph of some real analytic function g ( x ) over some open interval, then asbefore we can assign 0-cells to all strict inflection points and the two endpoints (one at thesingular point q and the other on the boundary of B ( q ) ). Next, we can assign 1-cells to allline segments in between these finitely many adjacent 0-cells. Then, we repeat this sameprocedure for each component inside B ( q ) \ { q } , obtaining a finite cell decomposition for X ∩ B ( q ) . (If there is no line component inside B ( q ) \ { q } , X ∩ B ( q ) is a single point at q .Assign a 0-cell at point q .)Finally, since X (by the compactness) can be covered by finitely many open sets in theform of either I × J around a non-singular point p or B ( q ) centered at a singular point q in S , X has a finite cell decomposition. It is easy to check that each cell in A is a semi-algebraic set. Indeed, any 1-cell is a continuous open line segment on the variety V ( f ) ,and so is the intersection of an open rectangle with V ( f ) , which is semi-algebraic. If γ isa shortest-length piecewise C -curve between two points in X , the intersection of γ witheach cell in A is either empty or consists of a single component that is either a singletonor a geodesic line segment. (cid:3) Theorem 2.3 assumes that X satisfies the compactness and connectedness properties,the next corollary shows that these two conditions are actually redundant. Corollary 2.8.
Suppose f ( x , y ) is an irreducible polynomial function and X = V ( f ) is theaffine algebraic variety determined by the zeros of f . Then X has a cell decomposition A with the desired finiteness property.Proof. It suffices to prove for the case when X is connected, but not necessarily compact.In general, if each connected component of X has a cell decomposition with the finitenessproperty, so does X . From now on, let us assume that X is connected.There exists a large positive integer N such that the open ball B ( , N ) centered at 0 withradius N contains all points of X for which either ∂ f ∂ x =
0, or ∂ f ∂ y =
0. Such an N exists,because of (2.3) and (2.4). For the part of X contained inside the closed ball ¯ B ( , N ) ,it can again be covered by finitely many open sets in the form of either I × J around anon-singular point p or B ( q ) for a singular point q . It guarantees the existence of a celldecomposition with the finiteness property using the same proof as Theorem 2.3.Next, for each n ≥ N , consider the closed annulus ¯ A ( n , n + ) centered at 0 with innerand outer radii being n and n +
1, respectively. Then the intersection of X with ¯ A ( n , n + ) can be covered by finitely many open sets in the form of only I × J , which also guaranteesthe existence of a cell decomposition with the finiteness property.Lastly, we combine the cell decomposition for X ∩ ¯ B ( , N ) with those for X ∩ ¯ A ( n , n + ) ,where n ≥ N , thus yielding a cell decomposition A with the desired finiteness property. (cid:3) Now we are ready for the following general theorem concerning an arbitrary affinealgebraic set in R . Theorem 2.9.
Suppose X is any arbitrary affine algebraic set in the plane, then X has acell decomposition A with the finiteness property. Furthermore, ( X , A ) is a CW complex.Proof. Since R is Noetherian, Hilbert Basis Theorem implies that R [ x , y ] is also Noether-ian. Then X = V ( f , . . . , f k ) for finitely many polynomials f i ∈ R [ x , y ] , where 1 ≤ i ≤ k .For each f i , we can write it as a product of finitely many irreducible polynomials, say f i , . . . , f m i i . Since V ( f , g ) = V ( f ) ∩ V ( g ) and V ( f · g ) = V ( f ) ∪ V ( g ) , X can be rewrittenas follows: X = k (cid:92) i = V ( f i ) ∪ . . . ∪ V ( f m i i ) . Distributing the intersections over the unions, it turns out that X is a finite union of sets inthe following form: V ( f j ) ∩ V ( f j ) ∩ . . . ∩ V ( f j k k ) , where 1 ≤ j i ≤ m i , for each i = , . . . , k . If k ≥
2, the above expression consists of at most finitely many points because of thealgebraic geometry theorem that we’ve utilized earlier [6]. Therefore X is either an emptyset or consists of finitely many points in R , so the theorem follows trivially.Next suppose that k =
1, then X = V ( f ) ∪ . . . ∪ V ( f m ) . If m =
1, we are done. Ifnot, for each j = , . . . , m , V ( f j ) has a cell decomposition A j with the finiteness propertybased on Corollary 2.8. Consider the following set T : T = { x ∈ X : x ∈ V ( f j ) ∩ V ( f j (cid:48) ) , where 1 ≤ j (cid:54) = j (cid:48) ≤ m } . Then T is finite. Adjust A j for each j by adding a 0-cell for each point of V ( f j ) that liesin T , and then including extra 1-cells if necessary. Call the new cell decomposition A (cid:48) j .Let A = A (cid:48) ∪ . . . A (cid:48) m . If γ is a shortest-length piecewise C -curve between two pointsin X , then γ = ( γ ∩ V ( f ) ) (cid:91) . . . (cid:91) ( γ ∩ V ( f m ) ) . By the compactness of γ , for each j = , . . . , m , γ ∩ V ( f j ) consists of finitely manycomponents, each of which is either a singleton or a shortest-length piecewise C -pathbetween its two endpoints. Therefore, there are at most finitely many components in theintersection of γ with each 1-cell in A , each of which is either a singleton or a geodesicline segment. Moreover, if there are more than one component, then at least one endpointof one of the components must lie in T , thus contradicting the fact that none of the 1-cellsin A contains a point in T . As a conclusion, γ intersects each 1-cell in A at most once,and the intersection must be a geodesic line segment.Since X is a Hausdorff space and A is a cell decomposition, ( X , A ) is a cell complex.Moreover, A is locally finite based on the construction in Corollary 2.8. The two additionalconditions (C) and (W) for a CW complex are automatic [5]. Therefore ( X , A ) is a CWcomplex. (cid:3) Remark . In the proof of Theorem 2.9, we may take A to be the union of A , . . . , A m ,directly. Then the intersection of γ with every 1-cell in this cell decomposition may consistof more than one component, each of which is either a singleton or a geodesic line seg-ment. As a consequence, this cell decomposition also works for the purpose of proving thetheorem. However, we have chosen to adjust A j , for each j , in order to obtain a nicer celldecomposition, as illustrated in the proof of Theorem 2.9. GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 11
Corollary 2.11.
Suppose X is a finite union of arbitrary affine algebraic sets in the plane,then X has a cell decomposition A with the finiteness property. Furthermore, ( X , A ) is aCW complex.Proof. Since V ( f ) ∩ V ( g ) = V ( f + g ) (communicated to me through Dr. Hardt), and V ( f ) ∪ V ( g ) = V ( f · g ) , there exists f such that X = V ( f ) , which is thus an affine algebraicset. Applying Theorem 2.9 gives the desired result. (cid:3)
3. T
HE STRATIFICATION OF AN OPEN SEMI - ALGEBRAIC SET IN R Suppose g ( x , y ) , . . . , g m ( x , y ) are nonzero polynomial functions in two real variables,and assume they are irreducible and distinct. Define the open semi-algebraic set Y asbelow:(3.1) Y = { g ( x , y ) > , . . . , g m ( x , y ) > } , where m ≥ . Given such a Y , consider the following affine algebraic set X :(3.2) X = V ( g ) ∪ . . . ∪ V ( g m ) , which is closed. Then R \ X is a disjoint union of connected open planar regions, in eachof which the value of g j is either entirely greater than 0 or less than 0, due to the continuityof g j for each j = , , . . . , m , and the connectedness. Therefore, Y consists of some of(possibly none) these connected open planar regions. It suffices to come up with a properstratification for each such individual open planar region, then a desired stratification for Y is thus obtained by taking the union.Since X is also equivalent to V ( g · · · g m ) , X has a cellular stratification with the finite-ness property based on Theorem 2.9. Let’s start with make an elementary observation. Lemma 3.1.
Let Y be given as in (3.1), and let P be a nonempty connected component ofY . If the boundary of P is nonempty, then it is contained in the affine algebraic set X asdefined in (3.2).Proof.
Suppose a is a boundary point of P , then a is not inside P otherwise it is an interiorpoint. There exists a sequence { a n } ∞ of points in P converging to a , which, by the conti-nuity of g j , implies that g j ( a n ) → g ( a ) for each j = , , . . . , m . Thus g j ( a ) ≥ j . If g j ( a ) = j , then we are done. If not, a is contained in one of theconnected components in Y other than P , making a an exterior point of P . (cid:3) From Lemma 3.1, if P is bounded, its boundary is nonempty (using the fact that R isconnected and unbounded), and thus is contained in X . We want to employ the stratificationof X to get a cell decomposition for P . One such strategy is to divide P using vertical stripswhose endpoints are determined by the 0-cells on the boundary of P . Example 3.2. let g = − y + ( x − )( x − ) and g = y − ( x − )( x + ) . Then Y is theconnected region bounded between two graphs as shown in Figure 2. The 0-cells on theupper boundary of Y determined by g are at points x = − , − , , ,
2; and the 0-cellson the lower boundary determined by g are at points x = − , ,
2. Projecting these 0-cells upon the x -axis partitions [ − , ] into five subintervals, each of which gives rise to avertical strip. It follows that we can divide Y into four sets: Y , . . . , Y , each of which has atop lying on the graph of g , a bottom that is on the graph of g , and two sides being eithera vertical line interval, or empty. We know how to construct a cell decomposition for eachof the Y i from our previous discussions, thus obtaining a cell decomposition for Y . Indeed,for each i = , . . . , Y i is a union of a region of type II and one or two open vertical lineintervals.In general , we can apply a similar idea to the open semi-algebraic set Y . Proposition 3.3.
Let Y be an open semi-algebraic set given as before, and P be a connectedcomponent of Y . Suppose that P is bounded, then P has a cell decomposition A with thefiniteness property. Furthermore, (P, A ) is a CW complex. GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 13 Y Y Y F IGURE
2. Divide Y through vertical strips determined by the 0-cells onthe boundary of Y . Proof.
By hypothesis, g , . . . , g m are irreducible and distinct, therefore the point of inter-section of V ( g i ) and V ( g j ) , for i (cid:54) = j , is at most a finite set. Include these points as 0-cellsin the cell decomposition for each V ( g i ) , i = , . . . , m . (If a new 0-cell is within a 1-cell,divide the 1-cell into two new 1-cells.) Indeed, since each cell decomposition for V ( g i ) exists by Theorem 2.9, a cell decomposition for X also exists by combining them. Call it A , then ( X , A ) is a CW complex because A is locally finite.We claim that the cell decomposition A satisfies the property that every 0- or 1-cell in A is either entirely contained in the boundary of P or entirely not. This is obviously truefor all the 0-cells. For the 1-cells, the proof is as follows. Let e be a 1-cell in A whichhas a nonempty intersection with the boundary of P . Without loss of generality, we mayassume that e belongs to the cell decomposition of V ( g ) . Consider the set S = { ( x , y ) ∈ e : ( x , y ) is on the boundary of P } , then S is closed and nonempty. It suffices to show that S is also open so that S is equal to e by the connectedness of e .Given ( x , y ) ∈ S , there exists an open neighborhood I × J around ( x , y ) such that { ( x , y ) ∈ I × J : g ( x , y ) = } is the graph of a real analytic function h over the x -axis (orthe y -axis). We may choose I × J to be so small that it doesn’t intersect V ( g ) , . . . , V ( g m )due to the fact that ( x , y ) is not an intersection point and so is at a positive distance fromeach of the closed sets V ( g j ) , where 2 ≤ j ≤ m . It follows that g j ( x , y ) is entirely > ( x , y ) ∈ I × J for each 2 ≤ j ≤ m , since g j ( x , y ) > j . On the other hand, the graphof h inside the open rectangle I × J divides it into two connected components, namely A = { ( x , y ) ∈ I × J : y < h ( x ) } and B = { ( x , y ) ∈ I × J : y > h ( x ) } . It is easy to check that g > P , say A . Then, A is a subset of P . As a result, everypoint in the set { ( x , y ) ∈ I × J : g ( x , y ) = } is a boundary point of P . Therefore S is alsoan open subset of e . Thus e belongs to the boundary of P and so does its closure. Since the boundary of P is a subset of X according to Lemma 3.1, it follows that the boundary of P is a finite subcomplex of X , because it is compact [5].Projecting the closure of P onto the x -axis, the image is a finite closed interval, say [ a , b ] ,where − ∞ < a < b < ∞ . Furthermore, projecting the 0-cells on the boundary of P divides [ a , b ] into finitely many intervals, say a = a < a < . . . < a n = b . For each i = , . . . , n − a i intersects the boundary of P at finitely many points.This is because g j ( a i , y ) = j = , . . . , m , unless g j ( a i , y ) ≡
0. If g j ( a i , y ) ≡ j , then the vertical line at a i divides the plane intotwo halves, so P lies inside only one of the two halves. Therefore, a ≥ a i or b ≤ a i , resultingin a contradiction. On the other hand, at x = a or x = b , the intersection of the vertical linewith the boundary of P is a finite disjoint union of closed vertical line intervals (of finitelengths) and isolated points. Indeed, the intersection is compact, so it consists of onlyfinitely many connected components, each of which is a connected subset of a real line.By the connectedness of the real line R , if a connected component has at least two points,it is an interval which is also closed and bounded in our case; if a connected componenthas only one point, then it is isolated from the others with respect to the subspace topologyinduced from R .For each i = , . . . , n −
1, add the finitely many points of intersection of the verticalline at a i and the boundary of P as 0-cells in the cell decomposition for the boundary of P . For i =
0, or n , we add these new 0-cells: the finitely many isolated points and the endpoints of the vertical line segments in the intersection of the vertical line at a or b withthe boundary of P . It follows that with respect to this new cell decomposition, each 1-cellon the boundary of P lies directly over one and only one open interval ( a i , a i + ) for some i = , . . . , n −
1, except for the possible vertical 1-cells at the two endpoints a , b and forthose 1-cells that are graphs over the y-axis.We want to modify these 1-cells which are graphs over the y-axis so that they becomegraphs over the x-axis as well. This can be done by dividing these 1-cells further. Let e be one of such 1-cells, and without loss of generality, we may assume that e is carriedby V ( g ) . Based on our construction in Theorem 2.3, there exists an open neighborhood I × J of e such that¯ e = { ( x , y ) ∈ ¯ I × ¯ J : g ( x , y ) = } = { ( h ( y ) , y ) : y ∈ ¯ J } , for some real analytic function h . If h is linear, it’s either constant, in which case we geta vertical line segment, or has a nonzero slope, in which case the graph of h as a function of y is also a graph over the x -axis. Suppose h is nonlinear, we can insert its local maximum points as 0-cells, which are finitely many according to Corollary 2.2. It follows that everynew 1-cell in e is either strictly increasing or decreasing over the y-axis, thus becominga graph over the x-axis as well. Furthermore, each such graph when viewed as over the x -axis is also real analytic, because given ( x , y ) ∈ e with h (cid:48) ( y ) (cid:54) = ∂ g ∂ x ( x , y ) (cid:54) = , by hypothesis, and moreover ∂ g ∂ x ( x , y ) · h (cid:48) ( y ) + ∂ g ∂ y ( x , y ) = = ⇒ ∂ g ∂ y ( x , y ) (cid:54) = . Repeat the previous process for these new 0-cells, that is, first project them onto the x -axis, then include as 0-cells for the points of intersection of the vertical lines with theboundary of P . In the end, we obtain a cell decomposition in which each 1-cell is eitheran open vertical interval or a real analytic function over ( a i , a i + ) for some i = , . . . , n − GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 15 (using the same notation for the new partition of [ a , b ] as before). Furthermore, thereare at least two separate non-vertical 1-cells lying over ( a i , a i + ) for all i , otherwise P isdisconnected or unbounded. A picture for the intersection of the closure of P with thevertical strip [ a i , a i + ] × R is shown in Figure 3. We note that the 1-cells lying over acommon interval might share common endpoints in their closures. (i) a middle intervala i a i+1 (ii) an end intervala a a i-1 The shaded area is in P, and its boundary is shown as dots. F IGURE
3. The part of P and its boundary lying over an interval [ a i , a i + i ] .For each ( a i , a i + ) , since only finitely many 1-cells spread out over it, the open verticalstrip ( a i , a i + ) × R subtracting these 1-cells consist of finitely many connected open regionsbounded by at least one of these 1-cells on the top or bottom, and by a vertical line intervalor a point on the two sides (see Figure 4). We call such a connected region a basic (open)region . (An exception will be discussed soon.) Then the intersection of P and the openvertical strip is a finite union of these bounded basic regions which intersect P nontrivially.It follows that P can be partitioned into finitely many basic regions together with someof the open vertical sides for these regions. Two examples have already been shown inFigure 2 and Figure 3.Since a basic region is a region of type II, a cell decomposition with the finitenessproperty exists [10]. More precisely, each basic region is a finite intersection of openpolynomial half planes (that is, an open region below the graph of a polynomial functionup to rotation, which is analogous to an open half plane); each polynomial half plane has acell decomposition with a sequence of 1-cells (in the same shape of the graph) convergingto the boundary; then overlaying the cell decomposition for each open polynomial halfplane leads to a cell decomposition for the basic region.For any vertical open interval, whenever it is included as a side of a basic region in P ,we don’t do anything with that side. That is to say, we exclude the cell decomposition ofan open polynomial half plane corresponding to that particular side when performing the topbottomside side topbottom side topbottom(i) (ii) (iii) F IGURE
4. A basic (open) region.overlapping. One can check that such a cell decomposition is actually locally finite. Since P is also Hausdorff, ( P , A ) is a CW complex.There is one exception that we need to discuss carefully. The cell decomposition for X in (3.2) may consist of isolated points. (For example, x + y = removed F IGURE
5. The cell decomposition of a basic region with an isolatedpoint removed on one of its two sides.even more so that the local finiteness property can be achieved. It suffices to look at thefollowing example as an illustration.
Example 3.4.
Let C = [ − , ] × [ , ] \ { ( , ) } be a cube in the plane with the originbeing removed, then we want to come up with a cell decomposition of C that is also a CWcomplex. Use a sequence of semicircles centered at ( , ) with radii 1 / n , n ≥
1, to divide C . It follows that such a cell decomposition is locally finite (see Figure 5).Furthermore, we won’t have a situation in which for the two 1-cells adjacent to theisolated point being removed, one is in P and the other is not in P . See Figure 6. If one GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 17
This vertical cell is in P. This vertical cell is not in P. F IGURE
6. It is impossible to have a basic region with an isolated pointremoved such that the two adjacent 1-cells are not simultaneously insideor outside P .1-cell is not in P , it must lie in X , because g ≥ , . . . , g m ≥ a or b , and so one of the g j = a or b .Therefore the other 1-cell is also not in P . On the other hand, suppose one of the two 1-cells is in P , then they must correspond to an a i (cid:54) = a , b . Since x = a i intersects the boundaryof P at finitely many points, the other cell must also be in P . Thus the situation in Figure 6will never happen.It is easy to check that each cell in A is a semi-algebraic set. Indeed, each 2-cell isbounded by finitely many 1-cells each of which is either a vertical/horizontal line interval,or inherits the same shape from one of the top and bottom graphs of a basic region, or isa semicircle. It follows that all these 1-cells can be determined by algebraic equations,therefore making the 2-cell a semi-algebraic set.Given a shortest-length curve γ between two points in P (if it exists), γ intersects theclosure of each 1-cell at most finitely many times. Indeed, the closure of each 1-cell isthe graph of some real analytic function. If γ intersects it infinitely many times, we canfind an accumulation point in the intersection. Since γ is locally a straight line ( P is anopen set), the 1-cell must be also linear thus leading to a contradiction. Therefore, each1-cell interacts γ at most finitely many times, thus so does every 2-cell. As a result, thecell decomposition A satisfies the desired finiteness property. This finishes the proof ofthe proposition. (cid:3) In the proposition, we assume that P is a bounded component of Y . In fact, this hypoth-esis can be removed by dividing the plane into cubes and focusing the cell decompositionin each cube. This idea is formally stated as follows: Proposition 3.5.
Suppose Y is an open semi-algebraic set defined as in (3.1), and let Pbe a connected component of Y . Then there exists a cell decomposition A for P such that ( P , A ) is a CW complex satisfying the finiteness property.Proof. If P is equal to the whole plane, then there is nothing to prove. Otherwise, theboundary of P is nonempty. Let’s pick a positive integer R large enough so that the foursides of the cube [ − R , R ] × [ − R , R ] intersects the boundary of P at mostly finitely many points. Indeed, if there were infinitely many intersection points, say on the side x = R ,then there exists 1 ≤ j ≤ m such that V ( g j ) ∩ { x = R } is an infinite set. Since both g j and x − R are irreducible, they must have a common factor implying that g j = c ( x − R ) for some nonzero constant c . In this situation, enlarging R fixes the problem. In fact, if g j = c ( x − a ) , or c ( y − a ) for some a ∈ R , where 1 ≤ j ≤ m , we require R to be bigger than a . Suppose the cube [ − R , R ] × [ − R , R ] intersects the boundary of P trivially, then the cubeis either entirely contained in P or entirely not, due to the connectedness property of thecube. In the case that it is completely inside P , a cell decomposition for it is shown inFigure 7. This is quite obvious. The cells compose of the four vertices, the interiors of thefour sides, and the interior of the cube.F IGURE
7. A cell decomposition of a cube in R .Suppose the cube [ − R , R ] × [ − R , R ] intersects the boundary of P nontrivially. Theboundary of P inside the cube contains finitely many 0-cells. Like in Proposition 3.3, weadd at most finitely many more 0-cells to ensure that every 1-cell is a graph of some realanalytic function over the x -axis or an open vertical interval. Moreover, we want to alsoinclude the following points as 0-cells: the four vertices of the cube, and the points of inter-section of the boundary of P and the boundary of the cube. Projecting these 0-cells onto the x -axis partitions [ − R , R ] into finitely many subintervals, say − R = a < a < . . . < a n = R .Over each such subinterval [ a i , a i + ] , the bounded vertical strip ( a i , a i + ) × [ − R , R ] is againbeing subdivided into finitely many connected components, each of whose interior is ei-ther contained completely inside P or not (see Figure 8). Each such an interior region is abasic (open) region as we’ve seen earlier. Therefore, the intersection of P with the closedcube [ − R , R ] × [ − R , R ] is a finite disjoint union of basic regions, and finitely many verticalor horizontal intervals which could be open, closed, or half-closed. It follows that a celldecomposition exists with the finiteness property. Indeed, if any vertical or horizontal openinterval is included as a side of a basic region in P , we do nothing with that side as seen inProposition 3.3. Moreover, if an endpoint of a vertical or horizontal open interval is inside P , that side must be entirely in P as well. Indeed, every vertical or horizontal open intervallies completely inside either P , or the boundary of P , or the exterior of P . Therefore, ifa side does not lie in P , then it is inside the complement of P , which is a closed set thushaving the two endpoints of the side as well. It follows that this observation guaranteesthat our cell decomposition satisfies the local finiteness property.Since the plane R is a countable union of the cube [ − R , R ] × [ − R , R ] and other cubesin the form of [ nR , ( n + ) R ] × [ mR , ( m + ) R ] , where at least one of n , m ≥ , or ≤ − P at mostly finitely many times. Applying the GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 19 -R Ra i a i+1The shaded areas are in P. -RR F IGURE
8. The vertical strip ( a i , a i + ) × [ − R , R ] and the 1-cells con-tained in it.similar argument as for [ − R , R ] × [ − R , R ] gives a cell decomposition for the intersection of P with each of these cubes. For two adjacent cubes, their common edge could get 0-cellsfrom both cell decompositions, and this is fine because there are only finitely many 0-cellsin total. Consequently, we obtain a cell decomposition A for P which fulfills the localfiniteness property, thus making ( P , A ) a CW complex. Furthermore, A also satisfies thefiniteness property. (cid:3) Theorem 3.6.
Suppose Y is an open semi-algebraic set defined as in (3.1), then there existsa cell decomposition A for Y such that ( Y , A ) is a CW complex satisfying the finitenessproperty.Proof. Since this is true for each connected component of Y according to the previousproposition, this is also true for Y . (cid:3) Next we need to look at the general case of taking a finite union of sets in the formof { g ( x , y ) > , . . . , g m ( x , y ) > } , in which the g j are not necessarily irreducible. Since g ( x , y ) · h ( x , y ) > g ( x , y ) > , h ( x , y ) > g ( x , y ) < , h ( x , y ) < − g ( x , y ) > , − h ( x , y ) > g j are indeed irreducible. -R R-2R 2R-R-2RR2R (0, 0) F IGURE
9. When P is unbounded, we may divide the plane into small cubes. Theorem 3.7.
Suppose Y is a finite union of open semi-algebraic sets defined as in (3.1),more precisely, let Y beY = { g ( x , y ) > , . . . , g m ( x , y ) > } ∪ { g (cid:48) ( x , y ) > , . . . , g (cid:48) m (cid:48) ( x , y ) > } ∪ . . . ∪ { g (cid:48)(cid:48) ( x , y ) > , . . . , g (cid:48)(cid:48) m (cid:48)(cid:48) ( x , y ) > } , where the union is finite, and the g j , g (cid:48) j , . . . , g (cid:48)(cid:48) j are all irreducible polynomials. Thenthere exists a cell decomposition A for Y such that ( Y , A ) is a CW complex satisfying thefiniteness property.Proof. The idea is similar as to the proof in Proposition 3.3, however, there is a slightimprovement as regard to the finiteness property.First, let’s again definite X to be the union of all boundaries as follows: X = { g = } , . . . , { g m = } ∪ { g (cid:48) = } , . . . , { g (cid:48) m (cid:48) = } ∪ . . . ∪ { g (cid:48)(cid:48) = } , . . . , { g (cid:48)(cid:48) m (cid:48)(cid:48) = } , for which we can find a CW decomposition according to Corollary 2.2.Second, look at each connected component P that is in the union Y . Then the boundary ∂ P of P is contained in X , and each 0- or 1- cell in X is either entirely contained in ∂ P ornot. Then chop P up as before into basic regions (Propositions 3.3 and 3.5). Observe thatif a 1-cell is not in Y , then its two endpoints are also not in Y , because the complement of Y is closed. Thus the finiteness property has no problem for the 1-cells. However, this nolonger holds for the 0-cells. We might have a 0-cell that is not in Y , but is adjacent to two1-cells which are in Y (For example, see Figure 10). In order for the finiteness property tobe satisfied near such a 0-cell, let’s first look at the following example. Example 3.8.
Suppose C = [ , ] × [ , ] \ { ( , ) } is the unit cube without one of itscorners at ( , ) , then we want to look for a cell decomposition for C that is also locallyfinite. In order to do so, let us first divide C into four small cubes. Next, for the lowerleft cube which contains ( , ) , let’s divide it further into another four small cubes. Then,pick the lower left cube which contains ( , ) , and divide it again. Continue this processinfinitely many times. This actually results in a cell decomposition of C that is also localfinite everywhere in C . A demonstration is shown in Figure 11. GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 21 x=-1 x=1y=-1y=1 F IGURE
10. When Y = { x > − , x < } ∪ { y > − , y < } , look at thebasic region in the middle, the four 0-cells at corners are not in Y , butthey are adjacent to 1-cells that are in Y . F IGURE
11. A CW decomposition of a unit cube minus one corner.Following the example, if two 1-cells that are in Y meet at a 0-cell that is not in Y ,and these two 1-cells are not in the same vertical line, we may apply an analogous celldecomposition as for [ , ] × [ , ] \ { ( , ) } . Suppose these two 1-cells are on the samevertical line, we return to the exceptional case as discussed before in which we employsemicircles to further divide up the basic region.There is one more situation that is actually ‘troublesome’. Previously we’ve seen inProposition 3.3, if Y is in the form of (3.1), it is impossible to have a basic region whosevertical sides consist of more than one 1-cells such that not all 1-cells are simultaneouslyinside Y (see Figure 12). However, if Y is a union of at least two sets in the form of (3.1),this circumstance might not longer be true. For example, let Y = x > ∪ y >
0, and consider the cell decomposition of [ − , ] × [ − , ] ∩ Y . (That is to say, R = Y while the other is not. Our previous technique of overlayingcell decompositions with respect to the 1-cells on the boundary of a basic region fails here.In this situation, let us insert an additional horizontal 1-cell at the 0-cell that connects two x=0y=0 These two 1-cells are not simultaneously in Y. F IGURE
12. If Y = x > ∪ y >
0, there is a basic region with its left sideconsisting of two 1-cells one of which is in Y while the other is not.1-cells one of which is in Y while the other is not. Suppose this open line interval iscompletely contained in the basic region (that is, intersecting the top and bottom graphsat most at a point on the other side), then we can divide our original basic region into twobasic regions, eliminating this ‘troublesome’ case.However, it is very likely that such a horizontal line interval has a nontrivial intersec-tion with the top or bottom graph of the region before even reaching the other side (seeFigure 13). Based on our construction, the top and bottom graphs belong to one of thefollowing four types:(1) strictly increasing, convex upward;(2) strictly decreasing, convex upward;(3) convex downward;(4) linear.This is because, previously only inflection points and local minimum points were consid-ered for the 0-cells, and it was sufficient. But here let us also include the local maximumpoints as 0-cells in the case of convex downward (Dr. Hardt communicated this to me).This will greatly simplify the argument, since the top and bottom graphs now belong toone of the following three types:(1) strictly increasing;(2) strictly decreasing;(3) constant.If the horizontal line interval intersects the top or bottom graph at a point which isnot on the other side, either the top graph is strictly decreasing, or the bottom graph is GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 23 strictly increasing. Furthermore, since the two graphs don’t intersect except possibly at theother side, the horizontal 1-cell at the connecting 0-cell intersects only one of the top andbottoms graphs at exactly one point. Therefore, we may introduce another vertical line
13 2 F IGURE
13. Divide a basic region into three basic regions using a hor-izontal 1-cell at the troublesome 0-cell and a vertical 1-cell at the inter-section point.interval at the intersection point, so that the original basic region can be divided into threebasic regions, each of which is no longer ‘troublesome’ (see Figure 13). If more than onesuch connecting 0-cells were present, we may perform the above procedure consecutivelyfor each one of them.As a conclusion, there exists a cell decomposition A of Y such that ( Y , A ) is a CWcomplex and satisfies the finiteness property. Indeed, the technique of dividing in the‘troublesome’ case makes sure that each cell is still semi-algebraic and the local finitenessproperty also holds. (cid:3) Remark . In Proposition 3.3, we’ve shown how to use semicircles to decompose a basicregion if it has a side with a removed 0-cell between two 1-cells. In fact, we may alsodivide the basic region using the technique above.
4. T
HE INTERSECTION OF AN ALGEBRAIC SET AND AN OPEN SEMI - ALGEBRAIC SET
Suppose f , . . . , f k , g , . . . , g m are nonzero real-valued polynomials in two variables, andassume they are irreducible and distinct. Consider the following semi-algebraic set Z :(4.1) Z = { f ( x , y ) = , . . . , f k ( x , y ) = } ∩ { g ( x , y ) > , . . . , g m ( x , y ) > } . If k ≥
2, the first f i ’s determine at most finitely many points, thus Z is a finite set. Nowlet’s assume that k =
1. Define B as follows:(4.2) B = { ( x , y ) ∈ R : f ( x , y ) = , g j ( x , y ) = j ∈ { , . . . , m }} . Since B is at most a finite set, we can include each point in B as a 0-cell to the cell de-composition of { f ( x , y ) = } as guaranteed by Corollary 2.8. It follows that V ( f ) \ B isa union of 0- and 1-cells. However, this is not yet a cell decomposition for some 1-cells’endpoints might be in B . To fix this problem, for each of these 1-cells which contain atleast one endpoint in the set B , we replace it with infinitely many 1-cells. The idea can bebest illustrated by looking at the following two examples. Example 4.1.
The open unit interval ( , ) may be decomposed as below:0-cells : n , − n (4.3) 1-cells : ( n + , n ) , ( − n , − n + ) , where n ≥ . And the half-closed interval ( , ] may be decomposed as below:0-cells : n , ( n + , n ) , ( , ) , where n ≥ . So if an endpoint of a 1-cell is removed, we consider a sequence of 0-cells convergingto the endpoint; and the parts between consecutive 0-cells determine the infinitely many1-cells. As a result, V ( f ) \ B becomes a cell complex. Furthermore, it is locally finite,thus is a CW complex.For the semi-algebraic set Z , it consists of the cells in V ( f ) \ B that are also in { g ( x , y ) > . . . , g m ( x , y ) > } . We note that if a 1-cell in V ( f ) \ B is in Z , then its two endpointsmust be in Z too, otherwise they are in B , which is a contradiction. Thus Z is a subcomplexof V ( f ) \ B . The finiteness property for Z is automatic. Let’s summarize our result in thefollowing proposition. Proposition 4.2.
Let Z be a semi-algebraic set as defined in (4.1), there exists a cell de-composition A for Z such that ( Z , A ) satisfies the finiteness property which is also a CWcomplex. Let’s take a finite union of these sets and see what happens.
Proposition 4.3.
Let Z be a finite union of semi-algebraic sets as defined in (4.1), thenthere exists a cell decomposition A for Z such that ( Z , A ) satisfies the finiteness propertywhich is also a CW complex.Proof. Suppose Z is defined as follows: Z = { f = , . . . , f k = , g > , . . . , g m > } ∪ { f (cid:48) = , . . . , f (cid:48) k (cid:48) = , g (cid:48) > , . . . , g (cid:48) m (cid:48) > }∪ . . . ∪ { f (cid:48)(cid:48) = , . . . , f (cid:48)(cid:48) k (cid:48)(cid:48) = , g (cid:48)(cid:48) > , . . . , g (cid:48)(cid:48) m (cid:48)(cid:48) > } , GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 25 where the union is finite. We know that when k ≥
2, there are at most finitely many points.Therefore the above expression can be reduced to the following: Z = { f = , g > , . . . , g m > } ∪ { f (cid:48) = , g (cid:48) > , . . . , g (cid:48) m (cid:48) > } (4.5) ∪ . . . ∪ { f (cid:48)(cid:48) = , g (cid:48)(cid:48) > , . . . , g (cid:48)(cid:48) m (cid:48)(cid:48) > } ∪ { finitely many points } . Consider the finite union X of the affine algebraic sets V ( f ) , V ( f (cid:48) ) , . . . , V ( f (cid:48)(cid:48) ) , togetherwith the finitely many points in (4.5), then X has a CW decomposition. Next consider theunion ˜ B of sets in the form of (4.2):˜ B = { f = , g j = , ≤ j ≤ m } ∪ { f (cid:48) = , g (cid:48) j (cid:48) = , ≤ j (cid:48) ≤ m (cid:48) }∪ . . . ∪ { f (cid:48)(cid:48) = , g (cid:48)(cid:48) j (cid:48)(cid:48) = , ≤ j (cid:48)(cid:48) ≤ m (cid:48)(cid:48) } . It follows that ˜ B is a finite set. Include the points in ˜ B as 0-cells to the cell decompositionof X . Call it A . Then remove the 0-cells from X that are in ˜ B , and are not in the union Z .Fix these 1-cells whose endpoints are removed as before. Thus X \ B ∩ Z c gets a new CWdecomposition. Call it A . Pick these 0- and 1-cells in A that are contained Z , it turns outthat Z is a subcomplex of A . Indeed, it suffices to show that if a 1-cell is contained in Z , itsendpoints are contained in Z too. Let’s go back to the first cell decomposition A . Givena 1-cell e ∈ A , without loss of generality, we may assume that e comes from { f = } .If its endpoints are not in ˜ B , we pass it directly to A . It follows that if e is in Z , then itstwo endpoints are also in Z . Now let us suppose that at least one endpoint of e is in ˜ B ,say p . On the one hand, if p is not in Z , then p is removed from A , and e is replaced byinfinitely many 1-cells, each of which returns to the previous case. On the other hand, if p is in Z , then we keep p in A and the end of e connecting to p remains intact. Thereforeif e is in Z , then p is automatically in Z . Repeating the same argument for 1-cells comingfrom { f (cid:48) = } , . . . , { f (cid:48)(cid:48) = } , yields the desired conclusion that Z is a subcomplex. Thefiniteness property for Z is easy to check. (cid:3) GENERAL CASE
In general, an arbitrary semi-algebraic set X in the plane can be described as: X = I (cid:91) i = J (cid:92) j = { ( x , y ) ∈ R : f i , j ( x , y ) = , g i , j ( x , y ) > } , where f i , j , g i , j are nonzero real-valued polynomials in two variables. We see that X isa finite union of sets in the form obtained by taking the intersection of an algebraic set(i.e. { f ( x , y ) = , . . . , f k ( x , y ) = } ) with an open semi-algebraic set (i.e. { g ( x , y ) > , . . . , g m ( x , y ) > } ). What’s more, we may assume that the f i , g j are irreducible becauseof the following observations: { f ( x , y ) · ˜ f ( x , y ) = } = { f ( x , y ) = } (cid:91) { ˜ f ( x , y ) = } ; { g ( x , y ) · ˜ g ( x , y ) > } = (cid:0) { g ( x , y ) > } ∩ { ˜ g ( x , y ) > } (cid:1) (cid:91) (cid:0) {− g ( x , y ) > } ∩ {− ˜ g ( x , y ) > } (cid:1) . From previous results, we’ve known how to construct a CW decomposition with thefiniteness property for each of the following three types of semi-algebraic sets: ( I ) { f ( x , y ) = , . . . , f k ( x , y ) = } , (Theorem 2.9) ( II ) { g ( x , y ) > , . . . , g m ( x , y ) > } , (Theorem 3.6) ( III ) { f ( x , y ) = , . . . , f k ( x , y ) = } ∩ { g ( x , y ) > , . . . , g m ( x , y ) > } , (Proposition 4.2) . Now we are ready to take their finite unions. First, let’s begin with finite unions of thesame type. We’ve discussed them already in previous sections. Namely, Corollary 2.2 fora finite union of type (I); Theorem 3.6 for a finite union of type (II); and Proposition 4.3for a finite union of type (III).Next, let’s take a finite union of exactly two different types: (I) + (II), (I) + (III), and(II) + (III).
Lemma 5.1 (I + II) . Suppose W is a finite union of sets in the form of (I) and (II), then Whas a cell decomposition A that satisfies the finiteness property. However, ( W , A ) is notnecessarily a CW complex.Proof. From hypothesis, W is in the following form:(5.1) W = { f = } ∪ . . . ∪ { f k = } ∪ { g > , . . . , g m > } ∪ . . . ∪ { g (cid:48) > , . . . , g (cid:48) m (cid:48) > } . Let X be defined as below: X = { f = } ∪ . . . ∪ { f k = } ∪ { g = } ∪ . . . ∪ { g m = } (5.2) ∪ . . . ∪ { g (cid:48) = } ∪ . . . ∪ { g (cid:48) m (cid:48) = } . As usual, X has a CW decomposition; moreover, we can divide W up into basic regions.The cell decomposition is almost the same as in Propositions 3.3 and 3.5, with only oneexception. First, assume a basic (open) region (without boundary) is contained in W . InProposition 3.5, we see that if a side does not lie in the semi-algebraic set Y , then itstwo endpoints also do not lie in Y , which is essential for the local finiteness property on theboundary of a basic region. However, such a nice observation fails for W here, in particularat corner points or at isolated removed points on vertical sides. A simple counterexampleis W = { x + y = } ∪ { y > } (see Figure 14). Thus it is possible to have 1-cell on theboundary of a basic region which is not in W but either of whose endpoints is in W . If sucha situation happens, for example, at a corner or at a removed isolated point on a vertical GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 27 (0, 0) F IGURE
14. For W = { x + y = } ∪ { y > } , the cell decompositionis not locally finite at the origin.side, we must include this endpoint as a 0-cell, causing the local finiteness property to faildefinitely.Second, assume a basic region is not contained in W . Then let’s look at the cells on itsboundary. A 1-cell is in W if and only if it is contained in V ( f ) ∪ . . . ∪ V ( f k ) , which isclosed. Therefore the two endpoints of the 1-cell are both contained in W , yielding a celldecomposition.Therefore, a cell decomposition exists for W for which the local finiteness propertymight fail. However, the finiteness property can be checked to still hold. (cid:3) Lemma 5.2 (I + III) . Suppose W is a finite union of sets in the form of (I) and (III), thenW has a cell decomposition A that satisfies the finiteness property. Moreover, ( W , A ) is aCW complex.Proof. The proof is analogous to that of Proposition 4.3. More precisely, W takes thefollowing form by hypothesis: W = { f = } ∪ . . . ∪ { f k = } ∪ { ˜ f = , g > , . . . , g m > }∪ . . . ∪ { ˜ f (cid:48) = , g (cid:48) > , . . . , g (cid:48) m (cid:48) > } ∪ { finitely many points } . Let X be defined as below: X = { f = } ∪ . . . ∪ { f k = } ∪ { ˜ f = } ∪ . . . ∪ { ˜ f (cid:48) = } ∪ { finitely many points } , which has a CW decomposition. Furthermore, add the following set ˜ B as 0-cells to the celldecomposition.˜ B = { ˜ f = , g j = , ≤ j ≤ m } ∪ . . . ∪ { ˜ f (cid:48) = , g (cid:48) j (cid:48) = , ≤ j (cid:48) ≤ m (cid:48) } . Then we remove the 0-cells that are in ˜ B and are not in W . Fixing these 1-cells whoseendpoints are removed as in (4.3) and (4.4), and selecting those carried by W yields a CWcomplex for W , which is a subcomplex of X \ ˜ B ∩ W c . It remains to check that if a 1-cell e is contained in W , then its endpoints are also contained in W . In our construction, e iscarried entirely by one of the following affine algebraic sets: { f = } , . . . , { f k = } , { ˜ f = } , . . . , { ˜ f (cid:48) = } . There are two cases.
Case 1: e is carried by { f = } ∪ . . . ∪ { f k = } , then its endpointsare automatic in W by closedness. Case 2: e is carried by { ˜ f = } ∪ . . . ∪ { ˜ f (cid:48) = } . Theendpoints are also contained in W , and the proof is analogous to that in Proposition 4.3. (cid:3) Lemma 5.3 (II + III) . Suppose W is a finite union of sets in the form of (II) and (III),then W has a cell decomposition A that satisfies the finiteness property. However, ( W , A ) might not be a CW complex.Proof. The hypothesis says that W can be written in the following form: W = { g > , . . . , g m > } ∪ . . . ∪ { g (cid:48) > , . . . , g (cid:48) m (cid:48) > } ∪{ f = , ˜ g > , . . . , ˜ g ˜ m > } ∪ . . . ∪ { f (cid:48) = , ˜ g (cid:48) > , . . . , ˜ g (cid:48) ˜ m (cid:48) > }∪ { finitely many points } . By Lemma 5.1, there exists a cell decomposition with the finiteness property for the fol-lowing set W (cid:48) : W (cid:48) = { g > , . . . , g m > } ∪ . . . ∪ { g (cid:48) > , . . . , g (cid:48) m (cid:48) > }∪ { f = } ∪ . . . ∪ { f (cid:48) = } ∪ { finitely many points } , where every point ( x , y ) in the plane corresponds to a real algebraic set such as { ( x − x ) + ( y − y ) = } . Furthermore, add the following set ˜ B as 0-cells to the above celldecomposition (before dividing W (cid:48) into basic regions):˜ B = { f = , ˜ g j = , ≤ j ≤ ˜ m } ∪ . . . ∪ { f (cid:48) = , ˜ g (cid:48) j (cid:48) = , ≤ j (cid:48) ≤ ˜ m (cid:48) } . Then we need to remove the 0- and 1-cells that are not W , in particular these lying in theunion { f = } ∪ . . . ∪ { f (cid:48) = } . Based on our construction, each of these cells is on theboundary of some basic region. There are two different cases. Case 1: the basic region belongs to W .
Without loss of generality, we may assumethat every side has no isolated 0-cells, otherwise dividing the region further by introducinghorizontal and vertical line intervals as shown in Theorem 3.6. It follows that every 0-cellis at the corner and every side is made up of only 1-cell. Thus removing a 0- or 1-cell won’taffect the cell decomposition too much, except for some minor adjusts. In fact, we’ve seenall possible boundary conditions already.
Case 2: the basic region does not belong to W .
Removing a 1-cell won’t affect anything.However, removing a 0-cell might cause a problem. Suppose e is 1-cell adjacent to this0-cell, and e is in W . If e is on the boundary of a basic region contained in W , we returnto case 1. Otherwise, we need to fix this 1-cell (in particular, the half with the 0-cell asan endpoint) by replacing it with infinitely many smaller 1-cells and 0-cells according to(4.4).As a result, there exists a cell decomposition A for W . It is easy to check that ( W , A ) satisfies the finiteness property. However, this cell decomposition is not necessarily locallyfiniteness for the same reason as shown in Lemma 5.1. (cid:3) Finally, we are ready to take a finite union of all three different types: (I) + (II) + (III).
Lemma 5.4 (I + II + III) . Suppose W is a finite union of sets in the form of (I), (II) and(III), then W has a cell decomposition A that satisfies the finiteness property. However, ( W , A ) might not be a CW complex.Proof. We may write W as follows: W = { f = } ∪ . . . ∪ { f k = } ∪ { g > , . . . , g m > } ∪ . . . ∪ { g (cid:48) > , . . . , g (cid:48) m (cid:48) > }∪ { ˜ f = , ˜ g > , . . . , ˜ g ˜ m > } ∪ . . . ∪ { ˜ f (cid:48) = , ˜ g (cid:48) > , . . . , ˜ g (cid:48) ˜ m (cid:48) > } . Then the rest of the proof is similar to that of Lemma 5.3. (cid:3)
GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 29
Theorem 5.5.
Given any semi-algebraic set in the plane, it has a cell decomposition withthe finiteness property.Proof.
Combine Lemmas 5.1, 5.2, 5.3, and 5.4. (cid:3)
6. C
ONCLUSION
In this paper, we find a semi-algebraic stratification A , in particular a cell decomposi-tion, for any arbitrary semi-algebraic set X in the plane. Moreover, A satisfies an analyticcondition concerning geodesics. More precisely, suppose A , B are two arbitrary points in X , and γ is a piecewise C curve from A to B lying entirely in X such that its length is theshortest among all possible such curves. Then the intersection of γ with every cell in A iseither empty or consists of finitely many components, each of which is either a singletonor a geodesic line segment.Furthermore, when X is in one of the following cases, ( X , A ) turns out to be a CWcomplex, because the cell decomposition is locally finite.(1) X is a finite union of sets in the form of { f = , . . . , f k = } ;(2) X is a finite union of sets in the form of { g > , . . . , g m > } ;(3) X is a finite union of sets in the form of { ˜ f = , . . . , ˜ f k = , ˜ g > , . . . , ˜ g m > } ;(4) X is a finite union of sets in the form of { f = , . . . , f k = } and { ˜ f = , . . . , ˜ f k = , ˜ g > , . . . , ˜ g m > } .The future questions may concern higher dimensional semi-algebraic sets, or semi-analytic sets, or sub-analytic sets, or triangulations, or even more complicated analyticalconditions such as Lipschitz conditions (that is to say, whether each 1-cell is the graph ofa Lipschitz function). GEODESIC STRATIFICATION OF TWO-DIMENSIONAL SEMI-ALGEBRAIC SETS 31 R EFERENCES1. S. Krantz,
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Elementary structure of real algebraic varieties , Ann. of Math. (2) (1957), 545-556.10. C. Yang, A triangulation of semi-algebraic sets concerning an analytical condition for shortest-length curves ,eprint arXiv:2011.14938, Nov. 2020. D EPARTMENT OF M ATHEMATICS , R
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