Capítulo 4. Análisis de datos 69
4.3. Análisis de resultados 72
In this subsection we will focus only on motivating the B´ezout-like bounds of Chap- ter6by considering an example application of the results proved in Chapter6to a problem considered by Safey El Din and Trebuchet in [13].
In the following we will frequently make use of the notion ofgeometric multiplicity
in the manner of Fulton [16, §1.5] and Fulton [15, §2.1]; we briefly describe this notion here. Let k be an algebraically closed field of characteristic zero. Let V
be a subvariety (or subscheme) of kn = Spec(k[x1, . . . ,xn]), i.e. a subvariety of dimensionnaffine space with coordinate ringk[x1, . . . ,xn]. LetWbe an irreducible component ofV. In this case the local ringOW,V is given by the localization of the coordinate ring ofV at the prime idealI(W), that is
OW,V = (k[x1, . . . ,xn]/I(V))I(W),
see [15,§2.1] or the proof of Lemma 6.3.3, and see §6.1for the definition ofOW,V in a more general setting.
We will write` OW,V
= `
OW,V OW,V
for thegeometric multiplicityofWinVwhere ` OW,Vis the length ofOW,V as anOW,V-module. Recall that a moduleMhas length
nif there is a composition series M0 = M % M1 % · · · % Mn = {0} and this is the shortest such series. For a set of points in affine space the notion of geometric multiplicity defined above reduces to the usual notion of the multiplicity of a point as we will see in the Example1.3.5.
V = V(f1, . . . , fn)in kn and W = (a1, . . . ,an)an isolated point in V we have ` OW,V=dimk (k[x1, . . . ,xn]/(f1, . . . , fn))P
,
(1.17)
where P=(x1−a1, . . . ,xn−an)is the prime ideal of the point W.
Consider the intersection of two curves f,g inC2 = Spec(C[x,y]), if p = (a,b)is
an isolated point in the intersection V = V(f,g)then ` Op,V =dimC(C[x,y]/(f,g))(x−a,y−b) .
If we take V to be the intersection of the curves y= x2and the x-axis y= 0we have one isolated point at p=(0,0)with geometric multiplicity
` Op,V = dimC C[x,y]/(x2−y,y) (x,y) = dimCC[x,y](x,y)/(x2−y,y) = dimCC[[x,y]]/(x2−y,y) = 2.
Here the basis ofC[[x,y]]/(x2−y,y)given by{1,x}whereC[[x,y]]denotes the ring
of formal power series. Note that we may replace the localizationC[x,y](x,y)by its completionC[[x,y]]in this case, see Fulton [15,§1.6].
The motivating example for the work in Chapter6 we consider here comes from a problem considered by Safey El Din and Trebuchet in [13] when developing an algorithm to compute at least one point in each connected component of a smooth real algebraic set.
Suppose we have an arbitrary collection of polynomials f1, . . . , fm ink[x1, . . . ,xn] withm < nand with deg(fi) ≤ Dfor allidefining an affine variety in An. Further suppose we wish to find the critical locus of V(f1, . . . , fm) using the method of Lagrange multipliers. To do this, the system we wish to consider is the following
collection of polynomials ink[x1, . . . ,xn,l1, . . . ,lm] Fj = fj if j≤m l1 ∂f1 ∂xj−m +· · ·+lm ∂fm ∂xj−m −1 if j=m+1 l1 ∂ f1 ∂xj−m +· · ·+lm ∂fm ∂xj−m ifm+2≤ j≤m+n . (1.18)
We may then calculate the critical locus by using an algorithm such as Giusti, Lecerf and Salvy [18] (or Lecerf [24]) to compute the variety V = V(F1, . . . ,Fn+m). Let
W1, . . . ,Wt be the irreducible components ofV. The algorithms of Giusti, Lecerf and Salvy [18] and of Lecerf [24] have known running time bounds that depend on the sum of the degrees of theWi weighted by multiplicity, that is the running time bounds depend on the quantityδgiven by
δ= X
`(OWi,V) deg(Wi).
Hence to give refined running time bounds on the time to compute the critical lo- cus ofV(f1, . . . , fm) using the method of Lagrange multipliers the problem we wish to consider is the following. Letting V ⊂ An+m be the affine variety defined by
V = V(F1, . . . ,Fn+m), how do we provide a refined bound on the degrees of the irre- ducible componentsW1, . . . ,WjofVwith multiplicity, i.e. a bound which is sharper than the usual B´ezout bound in this case? More specifically, if we homogenize to obtain the projective closureV ⊂Pn+mwe could then apply the usual B´ezout bound inPn+mto obtain δ= j X i=1 `(OWi,V) deg(Wi)≤D n+m. (1.19)
Our goal is to obtain a sharper bound than this by making use of the natural bi- projective structure of the variety associated to the system of polynomials in (1.18). In fact from Corollary6.3.8we have the following bound
δ= j X = `(OWi,V) deg(Wi)≤ n+m−1 n−1 ! Dn,
Corollary 6.3.8 follows from Theorem 6.2.1 which we prove in Chapter 6. Ad- ditionally ifV(F1, . . . ,Fm) is a complete intersection, Corollary6.3.9gives us the slightly sharper bound
δ= j X i=1 `(OWi,V) deg(Wi)≤ n n−m ! Dm(D−1)n−m.
We note that the bounds obtained from Corollary 6.3.8 and Corollary 6.3.9 are sharper (at least for large degree) than the bound obtained from the standard pro- jective B´ezout bound given in (1.19).
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Chapter 2
Computing Characteristics Classes
in Projective Space
The method to compute Chern-Schwartz-MacPherson classes described here is based on several known formulas due to Aluffi[1, 2], and on the notion of the projective degrees of a rational map as expressed in Harris [14]. The main result of this chapter is Theorem2.3.1which gives a method to compute projective degrees.
In particular, in this chapter, given the idealI defining a projective variety V inPn we will compute the pushforward toPn of both the Segre class ofV inPn and the Chern-Schwartz-MacPherson class of V (we abuse notation and denote the push- forwards toPn as s(V,
Pn) and cS M(V) respectively). FromcS M(V) we may imme- diately obtain the Euler characteristic of V, χ(V) using the well-known relation which states that χ(V) is equal to the degree of the zero dimensional component ofcS M(V). The algorithm described may be implemented either symbolically, with the computations relying on Gr¨obner bases calculations, or numerically using ho- motopy continuation.
We now give an example of the computation of the Segre class, thecS M class and the Euler characteristic for a singular projective variety. Note that since the variety
V considered in the example is singular the results cS M(V) andχ(V) could not be obtained with standard Chern class computations.
Example 2.0.6. Let V = V(I) be the subvariety of P4 defined by the ideal I = (4x3x2x4x1−x30x1,x0x1x3x4−x32x3)in k[x0,x1,x2,x3,x4]. Also let A∗(P4) Z[h]/(h5)
be the Chow ring ofP4.
Using Algorithm2.3.2with input I we obtain the Segre class
s(V,P4)=768h4−128h3+16h2 ∈A∗(P4).
Using Algorithm 2.3.3 with input I we obtain the Chern-Schwartz-MacPherson class
cS M(V)= 5h4+8h3+12h2∈A∗(P4)
and/or the Euler characteristicχ(V)=5.
The organization of the remainder of chapter is as follows. In Section2.1we state the problem we wish to consider and review the general definitions of thecS M class and the Chow ring. We also give several important results relating to the computa- tion ofcS M classes.
In Section 2.2 we briefly review relevant background on the projective degrees of a rational map and state known formulas which expresses the Segre class andcS M class in terms of these projective degrees. Also in Section2.2 we review previous algorithms for the computation of Segre andcS M classes. Specifically we review algorithms of Aluffi[2] and Eklund, Jost and Peterson [8] for the computation of Segre classes and we review algorithms of Aluffi[2] and Jost [17] for the computa- tion ofcS M classes. Additionally we explain the relationship between the residual degrees computed by Eklund, Jost and Peterson in [8] and the projective degrees in (2.16). In light of this relationship one could see Algorithm 2.3.2 and Algorithm 2.3.3as refinements of the algorithms of [8] and [17] respectively; however we note that these methods are developed using very different theoretical tools, and a priori there is no obvious relationship between them.
In Section2.3 we state and prove Theorem 2.3.1 which is the main result of this chapter and which gives a new formula for calculating the projective degrees of a rational map defined by a homogeneous ideal. In Algorithm 2.3.1 we show how
the result of Theorem 2.3.1 can be used to compute the projective degrees of a rational map using a computer algebra system. We then apply Algorithm 2.3.1to give Algorithm2.3.2 which computes the Segre class and Algorithm 2.3.3which computes thecS M class.
In Section 2.4 we discuss the performance of our algorithm to compute Segre classes (Algorithm 2.3.2) and our algorithm to compute thecS M class (Algorithm 2.3.3). Run time performance for Algorithm2.3.2is compared with previous algo- rithms of Aluffi[2] and of Eklund, Jost and Peterson [8] which also compute Segre classes. The results of the running time comparisons for Segre classes are summa- rized in Table 2.1; we see that our algorithm performs favourably in most cases. Run time performance for Algorithm 2.3.3is compared with previous algorithms of Aluffi [2] and of Jost [17] which also compute the cS M class and/or the Euler characteristic. We also compare the Macaulay2 [13] implementation of Algorithm 2.3.3to the Macaulay2 built in routine “euler” which calculates Hodge numbers to compute the Euler characteristic. In all cases Algorithm2.3.3performs favourably in comparison to other known algorithms. The results are summarized in Table 2.2.
The Macaulay2 [13] and Sage [24] implementations of our algorithm for computing
cS M classes, Euler characteristics and Segre classes of projective varieties can be found at https://github.com/Martin-Helmer/char-class-calc, see Ap- pendixA.1for a description of the syntax used by our “CharClassCalc” Macaulay2 package. The Macaulay2 [13] implementation is also available as part of the “Char- acteristicClasses” package in Macaulay2 version 1.7 and above and can be ac- cessed using the option “Algorithm=>ProjectiveDegree”, see the Macually2 docu- mentationhttp://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.7/sh are/doc/Macaulay2/CharacteristicClasses/html/for further details.
2.1
Problem and Setting
SupposeV is an arbitrary subscheme of a projective spacePn over an algebraically closed field of characteristic zero. The problem we wish to consider is that of devising an effective and practical algorithmic method to compute the Segre class
s(V,Pn) and the Chern-Schwartz-MacPherson classcS M(V) as elements of the Chow ring of Pn. An algorithm which computescS M(V) automatically give us the Euler characteristicχ(V), since this information is contained directly incS M(V).
In this section we review the definition of Chow groups and Chow rings; this is im- portant as we will represent characteristic classes as elements of some Chow ring. We also define the Chern-Schwartz-MacPherson class in a general setting and dis- cuss some important results relating to the computation of thecS M class. A general definition of the Segre class is given in§1.2.2 in the introduction, specifically see (1.3) above.