Math 137 Notes: Undergraduate Algebraic Geometry
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Homogeneous Polynomial Solutions to Constant Coefficient Pde’S
HOMOGENEOUS POLYNOMIAL SOLUTIONS TO CONSTANT COEFFICIENT PDE'S Bruce Reznick University of Illinois 1409 W. Green St. Urbana, IL 61801 [email protected] October 21, 1994 1. Introduction Suppose p is a homogeneous polynomial over a field K. Let p(D) be the differen- @ tial operator defined by replacing each occurence of xj in p by , defined formally @xj 2 in case K is not a subset of C. (The classical example is p(x1; · · · ; xn) = j xj , for which p(D) is the Laplacian r2.) In this paper we solve the equation p(DP)q = 0 for homogeneous polynomials q over K, under the restriction that K be an algebraically closed field of characteristic 0. (This restriction is mild, considering that the \natu- ral" field is C.) Our Main Theorem and its proof are the natural extensions of work by Sylvester, Clifford, Rosanes, Gundelfinger, Cartan, Maass and Helgason. The novelties in the presentation are the generalization of the base field and the removal of some restrictions on p. The proofs require only undergraduate mathematics and Hilbert's Nullstellensatz. A fringe benefit of the proof of the Main Theorem is an Expansion Theorem for homogeneous polynomials over R and C. This theorem is 2 2 2 trivial for linear forms, goes back at least to Gauss for p = x1 + x2 + x3, and has also been studied by Maass and Helgason. Main Theorem (See Theorem 4.1). Let K be an algebraically closed field of mj characteristic 0. Suppose p 2 K[x1; · · · ; xn] is homogeneous and p = j pj for distinct non-constant irreducible factors pj 2 K[x1; · · · ; xn]. -
Projective Geometry: a Short Introduction
Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g. -
HE WANG Abstract. a Mini-Course on Rational Homotopy Theory
RATIONAL HOMOTOPY THEORY HE WANG Abstract. A mini-course on rational homotopy theory. Contents 1. Introduction 2 2. Elementary homotopy theory 3 3. Spectral sequences 8 4. Postnikov towers and rational homotopy theory 16 5. Commutative differential graded algebras 21 6. Minimal models 25 7. Fundamental groups 34 References 36 2010 Mathematics Subject Classification. Primary 55P62 . 1 2 HE WANG 1. Introduction One of the goals of topology is to classify the topological spaces up to some equiva- lence relations, e.g., homeomorphic equivalence and homotopy equivalence (for algebraic topology). In algebraic topology, most of the time we will restrict to spaces which are homotopy equivalent to CW complexes. We have learned several algebraic invariants such as fundamental groups, homology groups, cohomology groups and cohomology rings. Using these algebraic invariants, we can seperate two non-homotopy equivalent spaces. Another powerful algebraic invariants are the higher homotopy groups. Whitehead the- orem shows that the functor of homotopy theory are power enough to determine when two CW complex are homotopy equivalent. A rational coefficient version of the homotopy theory has its own techniques and advan- tages: 1. fruitful algebraic structures. 2. easy to calculate. RATIONAL HOMOTOPY THEORY 3 2. Elementary homotopy theory 2.1. Higher homotopy groups. Let X be a connected CW-complex with a base point x0. Recall that the fundamental group π1(X; x0) = [(I;@I); (X; x0)] is the set of homotopy classes of maps from pair (I;@I) to (X; x0) with the product defined by composition of paths. Similarly, for each n ≥ 2, the higher homotopy group n n πn(X; x0) = [(I ;@I ); (X; x0)] n n is the set of homotopy classes of maps from pair (I ;@I ) to (X; x0) with the product defined by composition. -
CHAPTER 0 PRELIMINARY MATERIAL Paul
CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic preliminary notation, both mathematical and logistical. Sec- tion 2 describes what algebraic geometry is assumed of the reader, and gives a few conventions that will be assumed here. Section 3 gives a few more details on the field of definition of a variety. Section 4 does the same as Section 2 for number theory. The remaining sections of this chapter give slightly longer descriptions of some topics in algebraic geometry that will be needed: Kodaira’s lemma in Section 5, and descent in Section 6. x1. General notation The symbols Z , Q , R , and C stand for the ring of rational integers and the fields of rational numbers, real numbers, and complex numbers, respectively. The symbol N sig- nifies the natural numbers, which in this book start at zero: N = f0; 1; 2; 3;::: g . When it is necessary to refer to the positive integers, we use subscripts: Z>0 = f1; 2; 3;::: g . Similarly, R stands for the set of nonnegative real numbers, etc. ¸0 ` ` The set of extended real numbers is the set R := f¡1g R f1g . It carries the obvious ordering. ¯ ¯ If k is a field, then k denotes an algebraic closure of k . If ® 2 k , then Irr®;k(X) is the (unique) monic irreducible polynomial f 2 k[X] for which f(®) = 0 . Unless otherwise specified, the wording almost all will mean all but finitely many. -
Algebra Polynomial(Lecture-I)
Algebra Polynomial(Lecture-I) Dr. Amal Kumar Adak Department of Mathematics, G.D.College, Begusarai March 27, 2020 Algebra Dr. Amal Kumar Adak Definition Function: Any mathematical expression involving a quantity (say x) which can take different values, is called a function of that quantity. The quantity (x) is called the variable and the function of it is denoted by f (x) ; F (x) etc. Example:f (x) = x2 + 5x + 6 + x6 + x3 + 4x5. Algebra Dr. Amal Kumar Adak Polynomial Definition Polynomial: A polynomial in x over a real or complex field is an expression of the form n n−1 n−3 p (x) = a0x + a1x + a2x + ::: + an,where a0; a1;a2; :::; an are constants (free from x) may be real or complex and n is a positive integer. Example: (i)5x4 + 4x3 + 6x + 1,is a polynomial where a0 = 5; a1 = 4; a2 = 0; a3 = 6; a4 = 1a0 = 4; a1 = 4; 2 1 3 3 2 1 (ii)x + x + x + 2 is not a polynomial, since 3 ; 3 are not integers. (iii)x4 + x3 cos (x) + x2 + x + 1 is not a polynomial for the presence of the term cos (x) Algebra Dr. Amal Kumar Adak Definition Zero Polynomial: A polynomial in which all the coefficients a0; a1; a2; :::; an are zero is called a zero polynomial. Definition Complete and Incomplete polynomials: A polynomial with non-zero coefficients is called a complete polynomial. Other-wise it is incomplete. Example of a complete polynomial:x5 + 2x4 + 3x3 + 7x2 + 9x + 1. Example of an incomplete polynomial: x5 + 3x3 + 7x2 + 9x + 1. -
An Introduction to Complex Algebraic Geometry with Emphasis on The
AN INTRODUCTION TO COMPLEX ALGEBRAIC GEOMETRY WITH EMPHASIS ON THE THEORY OF SURFACES By Chris Peters Mathematisch Instituut der Rijksuniversiteit Leiden and Institut Fourier Grenoble i Preface These notes are based on courses given in the fall of 1992 at the University of Leiden and in the spring of 1993 at the University of Grenoble. These courses were meant to elucidate the Mori point of view on classification theory of algebraic surfaces as briefly alluded to in [P]. The material presented here consists of a more or less self-contained advanced course in complex algebraic geometry presupposing only some familiarity with the theory of algebraic curves or Riemann surfaces. But the goal, as in the lectures, is to understand the Enriques classification of surfaces from the point of view of Mori-theory. In my opininion any serious student in algebraic geometry should be acquainted as soon as possible with the yoga of coherent sheaves and so, after recalling the basic concepts in algebraic geometry, I have treated sheaves and their cohomology theory. This part culminated in Serre’s theorems about coherent sheaves on projective space. Having mastered these tools, the student can really start with surface theory, in particular with intersection theory of divisors on surfaces. The treatment given is algebraic, but the relation with the topological intersection theory is commented on briefly. A fuller discussion can be found in Appendix 2. Intersection theory then is applied immediately to rational surfaces. A basic tool from the modern point of view is Mori’s rationality theorem. The treatment for surfaces is elementary and I borrowed it from [Wi]. -
Classification on the Grassmannians
GEOMETRIC FRAMEWORK SOME EMPIRICAL RESULTS COMPRESSION ON G(k, n) CONCLUSIONS Classification on the Grassmannians: Theory and Applications Jen-Mei Chang Department of Mathematics and Statistics California State University, Long Beach [email protected] AMS Fall Western Section Meeting Special Session on Metric and Riemannian Methods in Shape Analysis October 9, 2010 at UCLA GEOMETRIC FRAMEWORK SOME EMPIRICAL RESULTS COMPRESSION ON G(k, n) CONCLUSIONS Outline 1 Geometric Framework Evolution of Classification Paradigms Grassmann Framework Grassmann Separability 2 Some Empirical Results Illumination Illumination + Low Resolutions 3 Compression on G(k, n) Motivations, Definitions, and Algorithms Karcher Compression for Face Recognition GEOMETRIC FRAMEWORK SOME EMPIRICAL RESULTS COMPRESSION ON G(k, n) CONCLUSIONS Architectures Historically Currently single-to-single subspace-to-subspace )2( )3( )1( d ( P,x ) d ( P,x ) d(P,x ) ... P G , , … , x )1( x )2( x( N) single-to-many many-to-many Image Set 1 … … G … … p * P , Grassmann Manifold Image Set 2 * X )1( X )2( … X ( N) q Project Eigenfaces GEOMETRIC FRAMEWORK SOME EMPIRICAL RESULTS COMPRESSION ON G(k, n) CONCLUSIONS Some Approaches • Single-to-Single 1 Euclidean distance of feature points. 2 Correlation. • Single-to-Many 1 Subspace method [Oja, 1983]. 2 Eigenfaces (a.k.a. Principal Component Analysis, KL-transform) [Sirovich & Kirby, 1987], [Turk & Pentland, 1991]. 3 Linear/Fisher Discriminate Analysis, Fisherfaces [Belhumeur et al., 1997]. 4 Kernel PCA [Yang et al., 2000]. GEOMETRIC FRAMEWORK SOME EMPIRICAL RESULTS COMPRESSION ON G(k, n) CONCLUSIONS Some Approaches • Many-to-Many 1 Tangent Space and Tangent Distance - Tangent Distance [Simard et al., 2001], Joint Manifold Distance [Fitzgibbon & Zisserman, 2003], Subspace Distance [Chang, 2004]. -
1 Affine Varieties
1 Affine Varieties We will begin following Kempf's Algebraic Varieties, and eventually will do things more like in Hartshorne. We will also use various sources for commutative algebra. What is algebraic geometry? Classically, it is the study of the zero sets of polynomials. We will now fix some notation. k will be some fixed algebraically closed field, any ring is commutative with identity, ring homomorphisms preserve identity, and a k-algebra is a ring R which contains k (i.e., we have a ring homomorphism ι : k ! R). P ⊆ R an ideal is prime iff R=P is an integral domain. Algebraic Sets n n We define affine n-space, A = k = f(a1; : : : ; an): ai 2 kg. n Any f = f(x1; : : : ; xn) 2 k[x1; : : : ; xn] defines a function f : A ! k : (a1; : : : ; an) 7! f(a1; : : : ; an). Exercise If f; g 2 k[x1; : : : ; xn] define the same function then f = g as polynomials. Definition 1.1 (Algebraic Sets). Let S ⊆ k[x1; : : : ; xn] be any subset. Then V (S) = fa 2 An : f(a) = 0 for all f 2 Sg. A subset of An is called algebraic if it is of this form. e.g., a point f(a1; : : : ; an)g = V (x1 − a1; : : : ; xn − an). Exercises 1. I = (S) is the ideal generated by S. Then V (S) = V (I). 2. I ⊆ J ) V (J) ⊆ V (I). P 3. V ([αIα) = V ( Iα) = \V (Iα). 4. V (I \ J) = V (I · J) = V (I) [ V (J). Definition 1.2 (Zariski Topology). We can define a topology on An by defining the closed subsets to be the algebraic subsets. -
ON the FIELD of DEFINITION of P-TORSION POINTS on ELLIPTIC CURVES OVER the RATIONALS
ON THE FIELD OF DEFINITION OF p-TORSION POINTS ON ELLIPTIC CURVES OVER THE RATIONALS ALVARO´ LOZANO-ROBLEDO Abstract. Let SQ(d) be the set of primes p for which there exists a number field K of degree ≤ d and an elliptic curve E=Q, such that the order of the torsion subgroup of E(K) is divisible by p. In this article we give bounds for the primes in the set SQ(d). In particular, we show that, if p ≥ 11, p 6= 13; 37, and p 2 SQ(d), then p ≤ 2d + 1. Moreover, we determine SQ(d) for all d ≤ 42, and give a conjectural formula for all d ≥ 1. If Serre’s uniformity problem is answered positively, then our conjectural formula is valid for all sufficiently large d. Under further assumptions on the non-cuspidal points on modular curves that parametrize those j-invariants associated to Cartan subgroups, the formula is valid for all d ≥ 1. 1. Introduction Let K be a number field of degree d ≥ 1 and let E=K be an elliptic curve. The Mordell-Weil theorem states that E(K), the set of K-rational points on E, can be given the structure of a finitely ∼ R generated abelian group. Thus, there is an integer R ≥ 0 such that E(K) = E(K)tors ⊕ Z and the torsion subgroup E(K)tors is finite. Here, we will focus on the order of E(K)tors. In particular, we are interested in the following question: if we fix d ≥ 1, what are the possible prime divisors of the order of E(K)tors, for E and K as above? Definition 1.1. -
4 Jul 2018 Is Spacetime As Physical As Is Space?
Is spacetime as physical as is space? Mayeul Arminjon Univ. Grenoble Alpes, CNRS, Grenoble INP, 3SR, F-38000 Grenoble, France Abstract Two questions are investigated by looking successively at classical me- chanics, special relativity, and relativistic gravity: first, how is space re- lated with spacetime? The proposed answer is that each given reference fluid, that is a congruence of reference trajectories, defines a physical space. The points of that space are formally defined to be the world lines of the congruence. That space can be endowed with a natural structure of 3-D differentiable manifold, thus giving rise to a simple notion of spatial tensor — namely, a tensor on the space manifold. The second question is: does the geometric structure of the spacetime determine the physics, in particular, does it determine its relativistic or preferred- frame character? We find that it does not, for different physics (either relativistic or not) may be defined on the same spacetime structure — and also, the same physics can be implemented on different spacetime structures. MSC: 70A05 [Mechanics of particles and systems: Axiomatics, founda- tions] 70B05 [Mechanics of particles and systems: Kinematics of a particle] 83A05 [Relativity and gravitational theory: Special relativity] 83D05 [Relativity and gravitational theory: Relativistic gravitational theories other than Einstein’s] Keywords: Affine space; classical mechanics; special relativity; relativis- arXiv:1807.01997v1 [gr-qc] 4 Jul 2018 tic gravity; reference fluid. 1 Introduction and Summary What is more “physical”: space or spacetime? Today, many physicists would vote for the second one. Whereas, still in 1905, spacetime was not a known concept. It has been since realized that, even in classical mechanics, space and time are coupled: already in the minimum sense that space does not exist 1 without time and vice-versa, and also through the Galileo transformation. -
The Projective Geometry of the Spacetime Yielded by Relativistic Positioning Systems and Relativistic Location Systems Jacques Rubin
The projective geometry of the spacetime yielded by relativistic positioning systems and relativistic location systems Jacques Rubin To cite this version: Jacques Rubin. The projective geometry of the spacetime yielded by relativistic positioning systems and relativistic location systems. 2014. hal-00945515 HAL Id: hal-00945515 https://hal.inria.fr/hal-00945515 Submitted on 12 Feb 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The projective geometry of the spacetime yielded by relativistic positioning systems and relativistic location systems Jacques L. Rubin (email: [email protected]) Université de Nice–Sophia Antipolis, UFR Sciences Institut du Non-Linéaire de Nice, UMR7335 1361 route des Lucioles, F-06560 Valbonne, France (Dated: February 12, 2014) As well accepted now, current positioning systems such as GPS, Galileo, Beidou, etc. are not primary, relativistic systems. Nevertheless, genuine, relativistic and primary positioning systems have been proposed recently by Bahder, Coll et al. and Rovelli to remedy such prior defects. These new designs all have in common an equivariant conformal geometry featuring, as the most basic ingredient, the spacetime geometry. In a first step, we show how this conformal aspect can be the four-dimensional projective part of a larger five-dimensional geometry. -
Affine and Euclidean Geometry Affine and Projective Geometry, E
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda CHAPTER II: AFFINE AND EUCLIDEAN GEOMETRY AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 1. AFFINE SPACE 1.1 Definition of affine space A real affine space is a triple (A; V; φ) where A is a set of points, V is a real vector space and φ: A × A −! V is a map verifying: 1. 8P 2 A and 8u 2 V there exists a unique Q 2 A such that φ(P; Q) = u: 2. φ(P; Q) + φ(Q; R) = φ(P; R) for everyP; Q; R 2 A. Notation. We will write φ(P; Q) = PQ. The elements contained on the set A are called points of A and we will say that V is the vector space associated to the affine space (A; V; φ). We define the dimension of the affine space (A; V; φ) as dim A = dim V: AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda Examples 1. Every vector space V is an affine space with associated vector space V . Indeed, in the triple (A; V; φ), A =V and the map φ is given by φ: A × A −! V; φ(u; v) = v − u: 2. According to the previous example, (R2; R2; φ) is an affine space of dimension 2, (R3; R3; φ) is an affine space of dimension 3. In general (Rn; Rn; φ) is an affine space of dimension n. 1.1.1 Properties of affine spaces Let (A; V; φ) be a real affine space. The following statemens hold: 1.