# INTRODUCTION TO TOPOLOGY OF REAL ALGEBRAIC VARIETIES

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 INTRODUCTION TO TOPOLOGY OF REAL ALGEBRAIC VARIETIES OLEG VIRO 1. The Early Topological Study of Real Algebraic Plane Curves 1.1. Basic Definitions and Problems. A curve (at least, an algebraic curve) is something more than just the set of points which belong to it. There are many ways to introduce algebraic curves. In the elementary situation of real plane projective curves the simplest and most convenient is the following definition, which at first glance seems to be overly algebraic. By a real projective algebraic plane curve 1 of degree m we mean a homogeneous real polynomial of degree m in three variables, considered up to constant factors. If a is such a polynomial, then the equation a(x 0, x 1, x 2 ) = 0 defines the set of real points of the curve in the real projective plane RP 2. We let RA denote the set of real points of the curve A. Following tradition, we shall also call this set a curve, avoiding this terminology only in cases where confusion could result. A point (x 0 : x 1 : x 2 ) RP 2 is called a (real) singular point of the curve A if (x 0, x 1, x 2 ) R 3 is a critical point of the polynomial a which defines the curve. The curve A is said to be (real) nonsingular if it has no real singular points. The set of real points of a nonsingular real projective plane curve is a smooth closed one-dimensional submanifold of the projective plane. In the topology of nonsingular real projective algebraic plane curves, as in other similar areas, the first natural questions that arise are classification problems. 1.1.A (Topological Classification Problem). Up to homeomorphism, what are the possible sets of real points of a nonsingular real projective algebraic plane curve of degree m? 1 Of course, the full designation is used only in formal situations. One normally adopts an abbreviated terminology. We shall say simply a curve in contexts where this will not lead to confusion. 1

2 2 OLEG VIRO 1.1.B (Isotopy Classification Problem). Up to homeomorphism, what are the possible pairs (RP 2, RA) where A is a nonsingular real projective algebraic plane curve of degree m? It is well known that the components of a closed one-dimensional manifold are homeomorphic to a circle, and the topological type of the manifold is determined by the number of components; thus, the first problem reduces to asking about the number of components of a curve of degree m. The answer to this question, which was found by Harnack [Har-76] in 1876, is described in Sections 1.6 and 1.8 below. The second problem has a more naive formulation as the question of how a nonsingular curve of degree m can be situated in RP 2. Here we are really talking about the isotopy classification, since any homeomorphism RP 2 RP 2 is isotopic to the identity map. At present the second problem has been solved only for m 7. The solution is completely elementary when m 5: it was known in the last century, and we shall give the result in this section. But before proceeding to an exposition of these earliest achievements in the study of the topology of real algebraic curves, we shall recall the isotopy classification of closed one-dimensional submanifolds of the projective plane Digression: the Topology of Closed One-Dimensional Submanifolds of the Projective Plane. For brevity, we shall refer to closed one-dimensional submanifolds of the projective plane as topological plane curves, or simply curves when there is no danger of confusion. A connected curve can be situated in RP 2 in two topologically distinct ways: two-sidedly, i.e., as the boundary of a disc in RP 2, and one-sidedly, i.e., as a projective line. A two-sided connected curve is called an oval. The complement of an oval in RP 2 has two components, one of which is homeomorphic to a disc and the other homeomorphic to a Möbius strip. The first is called the inside and the second is called the outside. The complement of a connected one-sided curve is homeomorphic to a disc. Any two one-sided connected curves intersect, since each of them realizes the nonzero element of the group H 1 (RP 2 ; Z 2 ), which has nonzero self-intersection. Hence, a topological plane curve has at most one onesided component. The existence of such a component can be expressed in terms of homology: it exists if and only if the curve represents a nonzero element of H 1 (RP 2 ; Z 2 ). If it exists, then we say that the whole curve is one-sided; otherwise, we say that the curve is two-sided. Two disjoint ovals can be situated in two topologically distinct ways: each may lie outside the other one i.e., each is in the outside component of the complement of the other or else they may form an injective

3 INTRODUCTION TO TOPOLOGY OF REAL ALGEBRAIC VARIETIES 3 Figure 1 pair, i.e., one of them is in the inside component of the complement of the other in that case, we say that the first is the inner oval of the pair and the second is the outer oval. In the latter case we also say that the outer oval of the pair envelopes the inner oval. A set of h ovals of a curve any two of which form an injective pair is called a nest of depth h. The pair (RP 2, X), where X is a topological plane curve, is determined up to homeomorphism by whether or not X has a one-sided component and by the relative location of each pair of ovals. We shall adopt the following notation to describe this. A curve consisting of a single oval will be denoted by the symbol 1. The empty curve will be denoted by 0. A one-sided connected curve will be denoted by J. If A is the symbol for a certain two-sided curve, then the curve obtained by adding a new oval which envelopes all of the other ovals will be denoted by 1 A. A curve which is a union of two disjoint curves A and B having the property that none of the ovals in one curve is contained in an oval of the other is denoted by A B. In addition, we use the following abbreviations: if A denotes a certain curve, and if a part of another curve has the form A A A, where A occurs n times, then we let n A denote A A. We further write n 1 simply as n. When depicting a topological plane curve one usually represents the projective plane either as a disc with opposite points of the boundary identified, or else as the compactification of R 2, i.e., one visualizes the curve as its preimage under either the projection D 2 RP 2 or the inclusion R 2 RP 2. In this book we shall use the second method. For example, 1.2 shows a curve corresponding to the symbol J

4 4 OLEG VIRO 1.3. Bézout s Prohibitions and the Harnack Inequality. The most elementary prohibitions, it seems, are the topological consequences of Bézout s theorem. In any case, these were the first prohibitions to be discovered. 1.3.A (Bézout s Theorem (see, for example, [Wal-50], [Sha-77])). Let A 1 and A 2 be nonsingular curves of degree m 1 and m 2. If the set RA 1 RA 2 is finite, then this set contains at most m 1 m 2 points. If, in addition, RA 1 and RA 2 are transversal to one another, then the number of points in the intersection RA 1 RA 2 is congruent to m 1 m 2 modulo B. Corollary (1). A nonsingular plane curve of degree m is onesided if and only if m is odd. In particular, a curve of odd degree is nonempty. In fact, in order for a nonsingular plane curve to be two-sided, i.e., to be homologous to zero mod 2, it is necessary and sufficient that its intersection number with the projective line be zero mod 2. By Bézout s theorem, this is equivalent to the degree being even. 1.3.C. Corollary (2). The number of ovals in the union of two nests of a nonsingular plane curve of degree m does not exceed m/2. In particular, a nest of a curve of degree m has depth at most m/2, and if a curve of degree m has a nest of depth [m/2], then it does not have any ovals not in the nest. To prove Corollary 2 it suffices to apply Bézout s theorem to the curve and to a line which passes through the insides of the smallest ovals in the nests. 1.3.D. Corollary (3). There can be no more than m ovals in a set of ovals which is contained in a union of 5 nests of a nonsingular plane curve of degree m and which does not contain an oval enveloping all of the other ovals of the set. To prove Corollary 3 it suffices to apply Bézout s theorem to the curve and to a conic which passes through the insides of the smallest ovals in the nests. One can give corollaries whose proofs use curves of higher degree than lines and conics (see Section 3.8). The most important of such results is Harnack s inequality. 1.3.E. Corollary (4 (Harnack Inequality [Har-76])). The number of components of a nonsingular plane curve of degree m is at most (m 1)(m 2)

5 INTRODUCTION TO TOPOLOGY OF REAL ALGEBRAIC VARIETIES 5 The derivation of Harnack Inequality from Bézout s theorem can be found in [Har-76], and also [Gud-74]. However, it is possible to prove Harnack Inequality without using Bézout s theorem; see, for example, [Gud-74], [Wil-78] and Section 3.2 below Curves of Degree 5. If m 5, then it is easy to see that the prohibitions in the previous subsection are satisfied only by the following isotopy types. m Table 1 Isotopy types of nonsingular plane curves of degree m 1 J 2 0, 1 3 J, J 1 4 0, 1, 2, 1 1, 3, 4 5 J, J 1, J 2, J 1 1, J 3, J 4, J 5, J 6 For m 3 the absence of other types follows from 1.3.B and 1.3.C; for m = 4 it follows from 1.3.B, 1.3.C and 1.3.D, or else from 1.3.B, 1.3.C and 1.3.E; and for m = 5 it follows from 1.3.B, 1.3.C and 1.3.E. It turns out that it is possible to realize all of the types in Table 1; hence, we have the following theorem. 1.4.A. Isotopy Classification of Nonsingular Real Plane Projective Curves of Degree 5. An isotopy class of topological plane curves contains a nonsingular curve of degree m 5 if and only if it occurs in the m-th row of Table 1. The curves of degree 2 are known to everyone. Both of the isotopy types of nonsingular curves of degree 3 can be realized by small perturbations of the union of a line and a conic which intersect in two real points (Figure 2). One can construct these perturbations by replacing the left side of the equation cl = 0 defining the union of the conic C and the line L by the polynomial cl + εl 1 l 2 l 3, where l i = 0, i = 1, 2, 3, are the equations of the lines shown in 2, and ε is a nonzero real number which is sufficiently small in absolute value. It will be left to the reader to prove that one in fact obtains the curves in Figure 2 as a result; alternatively, the reader can deduce this fact from the theorem in the next subsection.

6 6 OLEG VIRO R R R R R R R R Figure 2 Figure 3 The isotopy types of nonempty nonsingular curves of degree 4 can be realized in a similar way by small perturbations of a union of two conics which intersect in four real points (Figure 3). An empty curve of degree 4 can be defined, for example, by the equation x 4 0 +x 4 1 +x 4 2 = 0. All of the isotopy types of nonsingular curves of degree 5 can be realized by small perturbations of the union of two conics and a line, shown in Figure 4. For the isotopy classification of nonsingular curves of degree 6 it is no longer sufficient to use this type of construction, or even the prohibitions in the previous subsection. See Section 1.13 and?? The Classical Method of Constructing Nonsingular Plane Curves. All of the classical constructions of the topology of nonsingular plane curves are based on a single construction, which I will call

7 INTRODUCTION TO TOPOLOGY OF REAL ALGEBRAIC VARIETIES 7 Figure 4 classical small perturbation. Some special cases were given in the previous subsection. Here I will give a detailed description of the conditions under which it can be applied and the results. We say that a real singular point ξ = (ξ 0 : ξ 1 : ξ 2 ) of the curve A is an intersection point of two real transversal branches, or, more briefly, a crossing, 2 if the polynomial a defining the curve has matrix of second partial derivatives at the point (ξ 0, ξ 1, ξ 2 ) with both a positive and a negative eigenvalue, or, equivalently, if the point ξ is a nondegenerate critical point of index 1 of the functions {x RP 2 x i 0} R x a(x)/x i deg a for i with ξ i 0. By Morse lemma (see, e.g. [Mil-69]) in a neighborhood of such a point the curve looks like a union of two real lines. Conversely, if RA 1,...,RA k are nonsingular mutually transverse curves no three of which pass through the same point, then all of the singular points of the union RA 1 RA k (this is precisely the pairwise intersection points) are crossings. 1.5.A (Classical Small Perturbation Theorem (see Figure 5)). Let A be a plane curve of degree m all of whose singular points are crossings, and let B be a plane curve of degree m which does not pass through the singular points of A. Let U be a regular neighborhood of the curve RA in RP 2, represented as the union of a neighborhood U 0 of the set of singular points of A and a tubular neighborhood U 1 of the submanifold RA U 0 in RP 2 U 0. Then there exists a nonsingular plane curve X of degree m such that: 2 Sometimes other names are used. For example: a node, a point of type A 1 with two real branches, a nonisolated nondegenerate double point.

8 8 OLEG VIRO RB RAA R RA RX Figure 5 (1) RX U. (2) For each component V of U 0 there exists a homeomorphism hv D 1 D 1 such that h(ra V ) = D D 1 and h(rx V ) = {(x, y) D 1 D 1 xy = 1/2}. (3) RX U 0 is a section of the tubular fibration U 1 RA U 0. (4) RX {(x 0 : x 1 : x 2 ) RP 2 a(x 0, x 1, x 2 )b(x 0, x 1, x 2 ) 0}, where a and b are polynomials defining the curves A and B. (5) RX RA = RX RB = RA RB. (6) If p RA RB is a nonsingular point of B and RB is transversal to RA at this point, then RX is also transversal to RA at the point. There exists ε > 0 such that for any t (0, ε] the curve given by the polynomial a + tb satisfies all of the above requirements imposed on X. It follows from (1) (3) that for fixed A the isotopy type of the curve RX depends on which of two possible ways it behaves in a neighborhood of each of the crossings of the curve A, and this is determined by condition (4). Thus, conditions (1) (4) characterize the isotopy type of the curve RX. Conditions (4) (6) characterize its position relative to RA. We say that the curves defined by the polynomials a + tb with t (0, ε] are obtained by small perturbations of A directed to the curve B. It should be noted that the curves A and B do not determine the isotopy type of the perturbed curves: since both of the polynomials b and b determine the curve B, it follows that the polynomials a tb with small t > 0 also give small perturbations of A directed to B. But these curves are not isotopic to the curves given by a + tb (at least not in U), if the curve A actually has singularities.

9 INTRODUCTION TO TOPOLOGY OF REAL ALGEBRAIC VARIETIES 9 Proof. Proof of Theorem 1.5.A We set x t = a + tb. It is clear that for any t 0 the curve X t given by the polynomial x t satisfies conditions (5) and (6), and if t > 0 it satisfies (4). For small t we obviously have RX t U. Furthermore, if t is small, the curve RX t is nonsingular at the points of intersection RX t RB = RA RB, since the gradient of x t differs very little from the gradient of a when t is small, and the latter gradient is nonzero on RA RB (this is because, by assumption, B does not pass through the singular points of A). Outside RB the curve RX t is a level curve of the function a/b. On RA RB this level curve has critical points only at the singular points of RA, and these critical points are nondegenerate. Hence, for small t the behavior of RX t outside RB is described by the implicit function theorem and Morse Lemma (see, for example, [Mil-69]); in particular, for small t 0 this curve is nonsingular and satisfies conditions (2) and (3). Consequently, there exists ε > 0 such that for any t (0, ε] the curve RX t is nonsingular and satisfies (1) (6) Harnack Curves. In 1876, Harnack [Har-76] not only proved the inequality 1.3.E in Section 1.3, but also completed the topological classification of nonsingular plane curves by proving the following theorem. 1.6.A (Harnack Theorem). For any natural number m and any integer c satisfying the inequalities (1) 1 ( 1) m 2 c m2 3m + 4, 2 there exists a nonsingular plane curve of degree m consisting of c components. The inequality on the right in 1 is Harnack Inequality. The inequality on the left is part of Corollary 1 of Bézout s theorem (see Section 1.3.B). Thus, Harnack Theorem together with theorems 1.3.B and 1.3.E actually give a complete characterization of the set of topological types of nonsingular plane curves of degree m, i.e., they solve problem 1.1.A. Curves with the maximum number of components (i.e., with (m 2 3m + 4)/2 components, where m is the degree) are called M-curves. Curves of degree m which have (m 2 3m + 4)/2 a components are called (M a)-curves. We begin the proof of Theorem 1.6.A by establishing that the Harnack Inequality 1.3.B is best possible. 1.6.B. For any natural number m there exists an M-curve of degree m.

10 10 OLEG VIRO RL RA RB 5 5 RB 5 Figure 6 Proof. We shall actually construct a sequence of M-curves. At each step of the construction we add a line to the M-curve just constructed, and then give a slight perturbation to the union. We can begin the construction with a line or, as in Harnack s proof in [Har-76], with a circle. However, since we have already treated curves of degree 5 and constructed M-curves for those degrees (see Section 1.4), we shall begin by taking the M-curve of degree 5 that was constructed in Section 1.4, so that we can immediately proceed to curves that we have not encountered before. Recall that we obtained a degree 5 M-curve by perturbing the union of two conics and a line L. This perturbation can be done using various curves. For what follows it is essential that the auxiliary curve intersect L in five points which are outside the two conics. For example, let the auxiliary curve be a union of five lines which satisfies this condition (Figure 6). We let B 5 denote this union, and we let A 5 denote the M-curve of degree 5 that is obtained using B 5. We now construct a sequence of auxiliary curves B m for m > 5. We take B m to be a union of m lines which intersect L in m distinct points lying, for even m, in an arbitrary component of the set RL RB m 1 and for odd m in the component of RL RB m 1 containing RL RB m 2. We construct the M-curve A m of degree m using small perturbation of the union A m 1 L directed to B m. Suppose that the M-curve A m 1 of degree m 1 has already been constructed, and suppose that RA m 1 intersects RL transversally in the m 1 points of the intersection RL RB m 1 which lie in the same component of the curve RA m 1 and in the same order as on RL. It is not hard to see that, for one of the two possible directions of a small perturbation of A m 1 L directed to B m, the line RL and the component of RA m 1 that it intersects give

11 INTRODUCTION TO TOPOLOGY OF REAL ALGEBRAIC VARIETIES 11 RA 5 RB 6 RA 5 RA 6 RB 5 RB 6 RB 7 RA 7 RB 7 RA 7 Figure 7 m 1 components, while the other components of RA m 1, of which, by assumption, there are ((m 1) 2 3(m 1) + 4)/2 1 = (m 2 5m + 6)/2, are only slightly deformed so that the number of components of RA m remains equal to (m 2 5m + 6)/2 + m 1 = (m 2 3m + 4)/2. We have thus obtained an M-curve of degree m. This curve is transversal to RL, it intersects RL in RL RB m (see 1.5.A), and, since RL RB m is contained in one of the components of the set RL RB m 1, it follows that the intersection points of our curve with RL are all in the same component of the curve and are in the same order as on RL (Figure 7). The proof that the left inequality in 1 is best possible, i.e., that there is a curve with the minimum number of components, is much simpler. For example, we can take the curve given by the equation x m 0 + xm 1 + xm 2 = 0. Its set of real points is obviously empty when m is even, and when m is odd the set of real points is homeomorphic to

12 12 OLEG VIRO RP 1 (we can get such a homeomorphism onto RP 1, for example, by projection from the point (0 : 0 : 1)). By choosing the auxiliary curves B m in different ways in the construction of M-curves in the proof of Theorem 1.6.B, we can obtain curves with any intermediate number of components. However, to complete the proof of Theorem 1.6.A in this way would be rather tedious, even though it would not require any new ideas. We shall instead turn to a less explicit, but simpler and more conceptual method of proof, which is based on objects and phenomena not encountered above Digression: the Space of Real Projective Plane Curves. By the definition of real projective algebraic plane curves of degree m, they form a real projective space of dimension m(m + 3)/2. The homogeneous coordinates in this projective space are the coefficients of the polynomials defining the curves. We shall denote this space by the symbol RC m. Its only difference with the standard space RP m(m+3)/2 is the unusual numbering of the homogeneous coordinates. The point is that the coefficients of a homogeneous polynomial in three variables have a natural double indexing by the exponents of the monomials: a(x 0, x 1, x 2 ) = i,j 0 i+j ma ij x m i j 0 x i 1 xj 2. We let RNC m denote the subset of RC m corresponding to the real nonsingular curves. It is obviously open in RC m. Moreover, any nonsingular curve of degree m has a neighborhood in RNC m consisting of isotopic nonsingular curves. Namely, small changes in the coefficients of the polynomial defining the curve lead to polynomials which give smooth sections of a tubular fibration of the original curve. This is an easy consequence of the implicit function theorem; compare with 1.5.A, condition (3). Curves which belong to the same component of the space RNC m of nonsingular degree m curves are isotopic this follows from the fact that nonsingular curves which are close to one another are isotopic. A path in RNC m defines an isotopy in RP 2 of the set of real points of a curve. An isotopy obained in this way is made of sets of real points of of real points of curves of degree m. Such an isotopy is said to be rigid. This definition naturally gives rise to the following classification problem, which is every bit as classical as problems 1.1.Aand 1.1.B. 1.7.A (Rigid Isotopy Classification Problem). Classify the nonsingular curves of degree m up to rigid isotopy, i.e., study the partition of the space RNC m of nonsingular degree m curves into its components.

13 INTRODUCTION TO TOPOLOGY OF REAL ALGEBRAIC VARIETIES 13 Figure 8 If m 2, it is well known that the solution of this problem is identical to that of problem 1.1.B. Isotopy also implies rigid isotopy for curves of degree 3 and 4. This was known in the last century; however, we shall not discuss this further here, since it has little relevance to what follows. At present problem 1.7.A has been solved for m 6. Although this section is devoted to the early stages of the theory, I cannot resist commenting in some detail about a more recent result. In 1978, V. A. Rokhlin [Rok-78] discovered that for m 5 isotopy of nonsingular curves of degree m no longer implies rigid isotopy. The simplest example is given in Figure 8, which shows two curves of degree 5. They are obtained by slightly perturbing the very same curve in Figure 4 which is made up of two conics and a line. Rokhlin s original proof uses argument on complexification, it will be presented below, in Section??? Here, to prove that these curves are not rigid isotopic, we use more elementary arguements. Note that the first curve has an oval lying inside a triangle which does not intersect the one-sided component and which has its vertices inside the other three ovals, and the second curve does not have such an oval but under a rigid isotopy the oval cannot leave the triangle, since that would entail a violation of Bézout s theorem. We now examine the subset of RC m made up of real singular curves. It is clear that a curve of degree m has a singularity at (1 : 0 : 0) if and only if its polynomial has zero coefficients of the monomials x m 0, xm 1 0 x 1, x m 1 0 x 2. Thus, the set of real projective plane curves of degree m having a singularity at a particular point forms a subspace of codimension 3 in RC m. We now consider the space S of pairs of the form (p, C), where p RP 2, C RC m, and p is a singular point of the curve C. S is clearly an algebraic subvariety of the product RP 2 RC m. The restriction to S of the projection RP 2 RC m RP 2 is a locally trivial fibration whose fiber is the space of curves of degree m with

14 14 OLEG VIRO a singularity at the corresponding point, i.e., the fiber is a projective space of dimension m(m + 3)/2 3. Thus, S is a smooth manifold of dimension m(m+3)/2 1. The restriction S RC m of the projection RP 2 RC m RC m has as its image precisely the set of all real singular curves of degree m, i.e., RC m RNC m. We let RSC m denote this image. Since it is the image of a (m(m+3)/2 1)-dimensional manifold under smooth map, its dimension is at most m(m + 3)/2 1. On the other hand, its dimension is at least equal m(m+3)/2 1, since otherwise, as a subspace of codimension 2, it would not separate the space RC m, and all nonsingular curves of degree m would be isotopic. Using an argument similar to the proof that dim RSC m m(m + 3)/2 1, one can show that the set of curves having at least two singular points and the set of curves having a singular point where the matrix of second derivatives of the corresponding polynomial has rank 1, each has dimension at most m(m + 3)/2 2. Thus, the set RSC m has an open everywhere dense subset consisting of curves with only one singular point, which is a nondegenerate double point (meaning that at this point the matrix of second derivatives of the polynomial defining the curve has rank 2). This subset is called the principal part of the set RSC m. It is a smooth submanifold of codimension 1 in RC m. In fact, its preimage under the natural map S RC m is obviously an open everywhere dense subset in the manifold S, and the restriction of this map to the preimage is easily verified to be a one-to-one immersion, and even a smooth imbedding. There are two types of nondegenerate real points on a plane curve. We say that a nondegenerate real double point (ξ 0 : ξ 1 : ξ 2 ) on a curve A is solitary if the matrix of second partial derivatives of the polynomial defining A has either two nonnegative or two nonpositive eigenvalues at the point (ξ 0, ξ 1, ξ 2 ). A solitary nondegenerate double point of A is an isolated point of the set RA. In general, a singular point of A which is an isolated point of the set RA will be called a solitary real singular point. The other type of nondegenerate real double point is a crossing; crossings were discussed in Section 1.5 above. Corresponding to this division of the nondegenerate real double points into solitary points and crossings, we have a partition of the principal part of the set of real singular curves of degree m into two open sets. If a curve of degree m moves as a point of RC m along an arc which has a transversal intersection with the half of the principal part of the set of real singular curves consisting of curves with a solitary singular point, then the set of real points on this curve undergoes a Morse modification of index 0 or 2 (i.e., either the curve acquires a solitary double point, which then becomes a new oval, or else one of the ovals contracts

15 INTRODUCTION TO TOPOLOGY OF REAL ALGEBRAIC VARIETIES 15 to a point (a solitary nondegenerate double point) and disappears). In the case of a transversal intersection with the other half of the principal part of the set of real singular curves one has a Morse modification of index 1 (i.e., two arcs of the curve approach one another and merge, with a crossing at the point where they come together, and then immediately diverge in their modified form, as happens, for example, with the hyperbola in the family of affine curves of degree 2 given by the equation xy = t at the moment when t = 0). A line in RC m is called a (real) pencil of curves of degree m. If a and b are polynomials defining two curves of the pencil, then the other curves of the pencil are given by polynomials of the form λa + µb with λ, µ R 0. By the transversality theorem, the pencils which intersect the set of real singular curves only at points of the principal part and only transversally form an open everywhere dense subset of the set of all real pencils of curves of degree m End of the Proof of Theorem 1.6.A. In Section 1.6 it was shown that for any m there exist nonsingular curves of degree m with the minimum number (1 ( 1) m )/2 or with the maximum number (m 2 3m+4)/2 of components. Nonsingular curves which are isotopic to one another form an open set in the space RC m of real projective plane curves of degree m (see Section 1.7). Hence, there exists a real pencil of curves of degree m which connects a curve with minimum number of components to a curve with maximum number of components and which intersects the set of real singular curves only in its principal part and only transversally. As we move along this pencil from the curve with minimum number of components to the curve with maximum number of components, the curve only undergoes Morse modifications, each of which changes the number of components by at most 1. Consequently, this pencil includes nonsingular curves with an arbitrary intermediate number of components Isotopy Types of Harnack M-Curves. Harnack s construction of M-curves in [Har-76] differs from the construction in the proof of Theorem 1.6.B in that a conic, rather than a curve of degree 5, is used as the original curve. Figure 9 shows that the M-curves of degree 5 which are used in Harnack s construction [Har-76]. For m 6 Harnack s construction gives M-curves with the same isotopy types as in the construction in Section 1.6. In these constructions one obtains different isotopy types of M-curves depending on the choice of auxiliary curves (more precisely, depending on the relative location of the intersections RB m RL). Recall that in

### Easy reading on topology of real plane algebraic curves

Easy reading on topology of real plane algebraic curves Viatcheslav Kharlamov and Oleg Viro This is a shortened version of introduction to book Topological Properties of Real Plane Algebraic Curves by

### TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS

TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS OSAMU SAEKI Dedicated to Professor Yukio Matsumoto on the occasion of his 60th birthday Abstract. We classify singular fibers of C stable maps of orientable

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it

### FIBER PRODUCTS AND ZARISKI SHEAVES

FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also

### Singular fibers of stable maps and signatures of 4 manifolds

359 399 359 arxiv version: fonts, pagination and layout may vary from GT published version Singular fibers of stable maps and signatures of 4 manifolds OSAMU SAEKI TAKAHIRO YAMAMOTO We show that for a

### Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### Row Ideals and Fibers of Morphisms

Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

### The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

### ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### Math 3000 Running Glossary

Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (

### POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).

### Math 231b Lecture 35. G. Quick

Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### arxiv:0908.3127v2 [math.gt] 6 Sep 2009

MINIMAL SETS OF REIDEMEISTER MOVES arxiv:0908.3127v2 [math.gt] 6 Sep 2009 MICHAEL POLYAK Abstract. It is well known that any two diagrams representing the same oriented link are related by a finite sequence

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

### 2.5 Gaussian Elimination

page 150 150 CHAPTER 2 Matrices and Systems of Linear Equations 37 10 the linear algebra package of Maple, the three elementary 20 23 1 row operations are 12 1 swaprow(a,i,j): permute rows i and j 3 3

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

### Math 225A, Differential Topology: Homework 3

Math 225A, Differential Topology: Homework 3 Ian Coley October 17, 2013 Problem 1.4.7. Suppose that y is a regular value of f : X Y, where X is compact and dim X = dim Y. Show that f 1 (y) is a finite

### Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

### CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

### FIXED POINT SETS OF FIBER-PRESERVING MAPS

FIXED POINT SETS OF FIBER-PRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 e-mail: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### Surface bundles over S 1, the Thurston norm, and the Whitehead link

Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3-manifold can fiber over the circle. In

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### Fiber sums of genus 2 Lefschetz fibrations

Proceedings of 9 th Gökova Geometry-Topology Conference, pp, 1 10 Fiber sums of genus 2 Lefschetz fibrations Denis Auroux Abstract. Using the recent results of Siebert and Tian about the holomorphicity

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points

Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a

### NOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004

NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σ-algebras 7 1.1 σ-algebras............................... 7 1.2 Generated σ-algebras.........................

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### SECTION 10-2 Mathematical Induction

73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

### Quadratic Equations in Finite Fields of Characteristic 2

Quadratic Equations in Finite Fields of Characteristic 2 Klaus Pommerening May 2000 english version February 2012 Quadratic equations over fields of characteristic 2 are solved by the well known quadratic

### arxiv: v1 [math.mg] 6 May 2014

DEFINING RELATIONS FOR REFLECTIONS. I OLEG VIRO arxiv:1405.1460v1 [math.mg] 6 May 2014 Stony Brook University, NY, USA; PDMI, St. Petersburg, Russia Abstract. An idea to present a classical Lie group of

### Axiom A.1. Lines, planes and space are sets of points. Space contains all points.

73 Appendix A.1 Basic Notions We take the terms point, line, plane, and space as undefined. We also use the concept of a set and a subset, belongs to or is an element of a set. In a formal axiomatic approach

### CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction

Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves

### RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific

### It is not immediately obvious that this should even give an integer. Since 1 < 1 5

Math 163 - Introductory Seminar Lehigh University Spring 8 Notes on Fibonacci numbers, binomial coefficients and mathematical induction These are mostly notes from a previous class and thus include some

### Mathematical Induction

Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural

### ON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE

Illinois Journal of Mathematics Volume 46, Number 1, Spring 2002, Pages 145 153 S 0019-2082 ON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE J. HUISMAN Abstract. Let C be a

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

### Subsets of Euclidean domains possessing a unique division algorithm

Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness

### CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

### Chapter 2 Limits Functions and Sequences sequence sequence Example

Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students

### Practice Problems for First Test

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.-

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### Students in their first advanced mathematics classes are often surprised

CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions

### Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

### TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS

TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms

### NONMEASURABLE ALGEBRAIC SUMS OF SETS OF REALS

NONMEASURABLE ALGEBRAIC SUMS OF SETS OF REALS MARCIN KYSIAK Abstract. We present a theorem which generalizes some known theorems on the existence of nonmeasurable (in various senses) sets of the form X+Y.

### Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### 10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

### Section 6-2 Mathematical Induction

6- Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that

### 1 Local Brouwer degree

1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.

### Review for Final Exam

Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters

### Approximation des courbes sur les surfaces rationnelles réelles

Approximation des courbes sur les surfaces rationnelles réelles Frédéric Mangolte (Angers) en collaboration avec János Kollár (Princeton) Poitiers, 14 novembre 2013 Frédéric Mangolte (Angers) Courbes et

### On The Existence Of Flips

On The Existence Of Flips Hacon and McKernan s paper, arxiv alg-geom/0507597 Brian Lehmann, February 2007 1 Introduction References: Hacon and McKernan s paper, as above. Kollár and Mori, Birational Geometry

### The Mathematics of Origami

The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................