Hardy–Littlewood circle method
In mathematics, the Hardy–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy and J. E. Littlewood, who developed it in a series of papers on Waring's problem; the initial idea is attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function. It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation without changing the broad lines. Hundreds of papers followed, as of 2013 the method still yields results; the method is the subject of a monograph Vaughan by R. C. Vaughan; the goal is to prove asymptotic behavior of a series: to show. This is done by taking the generating function of the series computing the residues about zero. Technically, the generating function is scaled to have radius of convergence 1, so it has singularities on the unit circle – thus one cannot take the contour integral over the unit circle; the circle method is how to compute these residues, by partitioning the circle into minor arcs and major arcs, bounding the behavior on the minor arcs.
The key insight is that, in many cases of interest, the singularities occur at the roots of unity, the significance of the singularities is in the order of the Farey sequence. Thus one can investigate the most significant singularities, and, if fortunate, compute the integrals; the circle in question was the unit circle in the complex plane. Assuming the problem had first been formulated in the terms that for a sequence of complex numbers an, n = 0, 1, 2, 3...we want some asymptotic information of the type an ~ Fwhere we have some heuristic reason to guess the form taken by F, we write f = ∑ a n z n a power series generating function. The interesting cases are where f is of radius of convergence equal to 1, we suppose that the problem as posed has been modified to present this situation. From that formulation, it follows directly from the residue theorem that I n = ∫ f z − d z = 2 π i a n for integers n ≥ 0, where the integral is taken over the circle of radius r and centred at 0, for any r with 0 < r < 1.
That is, this is a contour integral, with the contour being the circle described traversed once anti-clockwise. So far, this is elementary. We would like to take r = 1 directly. In the complex analysis formulation this is problematic, since the values of f are not in general defined there; the problem addressed by the circle method is to force the issue of taking r = 1, by a good understanding of the nature of the singularities f exhibits on the unit circle. The fundamental insight is the role played by the Farey sequence of rational numbers, or equivalently by the roots of unity ζ = exp . Here the denominator s, assuming that r/s is in lowest terms, turns out to determine the relative importance of the singular behaviour of typical f near ζ; the Hardy–Littlewood circle method, for the complex-analytic formulation, can be thus expressed. The contributions to the evaluation of In, as r → 1, should be treated in two ways, traditionally called major arcs and minor arcs. We divide the roots of unity ζ into two classes, according to whether s ≤ N, or s > N, where N is a function of n, ours to choose conveniently.
The integral In is divided up into integrals each on some arc of the circle, adjacent to ζ, of length a function of s. The arcs make up the whole circle; the sum of the integrals over the minor arcs is to be replaced by an upper bound, smaller in order than F. Stated boldly like this, it is not at all clear; the insights involved are quite deep. One clear source is the theory of theta functions. In the context of Waring's problem, powers of theta functions are the generating functions for the sum of squares function, their analytic behaviour is known in much more accurate detail for example. It is the case, as the false-colour diagram indicates, that for a theta function the'most important' point on the boundary circle is at z = 1. After that it is − i that matter most. While nothing in this guarantees that the analytical method will work, it does explain the rationale of using a Farey series-type criterion on roots of unity. In the case of Waring's problem, one takes a sufficiently high power of the generating function to force the situation in which the singularities, organised into the so-called singular series, predominate.
The less wasteful the estimates used on the rest, the finer the results. As Bryan Birch has put it, the method is inherently wasteful. That
A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single quaternary cubic polynomial, homogeneous of degree 3. Cubic surfaces are del Pezzo surfaces. If P 3 has homogeneous coordinates the set of points where X 3 + Y 3 + Z 3 + W 3 = 0 is a cubic surface called the Fermat cubic surface; the Clebsch surface is the set of points where X 3 + Y 3 + Z 3 + W 3 = 3 Cayley's nodal cubic surface is the set of points where W X Y + X Y Z + Y Z W + Z W X = 0 The Cayley–Salmon theorem states that a smooth cubic surface over an algebraically closed field contains 27 straight lines. These can be characterized independently of the embedding into projective space as the rational lines with self-intersection number −1, or in other words the −1-curves on the surface. An Eckardt point is a point. A smooth cubic surface can be described as a rational surface obtained by blowing up six points in the projective plane in general position.
The 27 lines are the exceptional divisors above the 6 blown up points, the proper transforms of the 15 lines in P 2 which join two of the blown up points, the proper transforms of the 6 conics in P 2 which contain all but one of the blown up points. Clebsch gave a model of a cubic surface, called the Clebsch diagonal surface, where all the 27 lines are defined over the field Q, in particular are all real; the 27 lines can be identified with some objects arising in representation theory. In particular, these 27 lines can be identified with 27 vectors in the dual of the E6 lattice so their configuration is acted on by the Weyl group of E6. In particular they form a basis of the 27-dimensional fundamental representation of the group E6; the 27 lines contain 36 copies of the Schläfli double six configuration. The 27 lines can be identified with the 27 possible charges of M-theory on a six-dimensional torus and the group E6 naturally acts as the U-duality group; this map between del Pezzo surfaces and M-theory on tori is known as mysterious duality.
There are other ways of thinking of these 27 lines. For example, if one projects the cubic from a point, not on any line we obtain a double cover of the plane branched along a smooth quartic curve; the 27 lines are mapped to 27 out of the 28 bitangents to this quartic curve. These two objects, together with the 120 tritangent planes of a canonic sextic curve of genus 4, form a "trinity" in the sense of Vladimir Arnold a form of McKay correspondence, can be related to many further objects, including E7 and E8, as discussed at trinities. An example of a singular cubic is Cayley's nodal cubic surface W X Y + X Y Z + Y Z W + Z W X = 0 with 4 nodal singular points at and its permutations. Singular cubic surfaces contain rational lines, the number and arrangement of the lines is related to the type of the singularity; the singular cubic surfaces were classified by Schlafli, his classification was described by Cayley and Bruce & Wall Bruce, J. W.. "On the classification of cubic surfaces", Journal of the London Mathematical Society, 19: 245–256, doi:10.1112/jlms/s2-19.2.245, ISSN 0024-6107, MR 0533323 Cayley, A.
"On the triple tangent planes of surfaces of the third order", Cambridge and Dublin Math. J. 4: 118–138 Cayley, Arthur, "A Memoir on Cubic Surfaces", Philosophical Transactions of the Royal Society of London, The Royal Society, 159: 231–326, doi:10.1098/rstl.1869.0010, ISSN 0080-4614, JSTOR 108997 Dolgachev, Classical Algebraic Geometry:a modern view, Cambridge University Press, doi:10.1017/CBO9781139084437, MR 2964027 Dolgachev, Igor V. "Luigi Cremona and cubic surfaces", Luigi Cremona, Incontro di Studio, 36, Istituto Lombardo di Scienze e Lettere, Milan, pp. 55–70, MR 2305952 Henderson, The twenty-seven lines upon the cubic surface, Reprinting of Cambridge Tracts in Mathematics and Mathematical Physics, No. 13, Merchant books, ISBN 978-1-60386-066-6, MR 0119139 Hunt, The geometry of some special arithmetic quotients, Lecture Notes in Mathematics, 1637, New York: Sprin
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques from commutative algebra, for solving geometrical problems about these sets of zeros; the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, parabolas, hyperbolas, cubic curves like elliptic curves, quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis and number theory. A study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, it becomes more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution. In the 20th century, algebraic geometry split into several subareas; the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more to the points with coordinates in an algebraically closed field. Real algebraic geometry is the study of the real points of an algebraic variety. Diophantine geometry and, more arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, p-adic fields.
A large part of singularity theory is devoted to the singularities of algebraic varieties. Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers, it consists of algorithm design and software development for the study of properties of explicitly given algebraic varieties. Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way, similar to its use in the study of differential and analytic manifolds; this is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring.
This means that a point of such a scheme may be either a subvariety. This approach enables a unification of the language and the tools of classical algebraic geometry concerned with complex points, of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach. In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 − 1 = 0. A "slanted" circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 − 1 = 0, x + y + z = 0. First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed.
We consider the affine space of dimension n over denoted An. When one fixes a coordinate system, one may identify An with kn; the purpose of not working with kn is to emphasize that one "forgets" the vector space structure that kn carries. A function f: An → A1 is said to be polynomial if it can be written as a polynomial, that is, if there is a polynomial p in k such that f = p for every point M with coordinates in An; the property of a function to be polynomial does not depend on the choice of a coordinate system in An. When a coordinate system is chosen, the regular functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k. Therefore, the set of the
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when V = R2 and V = R3 are the real projective line and the real projective plane where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, R3 denotes ordered triplets of real numbers; the idea of a projective space relates to perspective, more to the way an eye or a camera projects a 3D scene to a 2D image. All points that lie on a projection line, intersecting with the entrance pupil of the camera, are projected onto a common image point. In this case, the vector space is R3 with the camera entrance pupil at the origin, the projective space corresponds to the image points. Projective spaces can be studied as a separate field in mathematics, but are used in various applied fields, geometry in particular. Geometric objects, such as points, lines, or planes, can be given a representation as elements in projective spaces based on homogeneous coordinates.
As a result, various relations between these objects can be described in a simpler way than is possible without homogeneous coordinates. Furthermore, various statements in geometry can be made more consistent and without exceptions. For example, in the standard Euclidean geometry for the plane, two lines always intersect at a point except when parallel. In a projective representation of lines and points, such an intersection point exists for parallel lines, it can be computed in the same way as other intersection points. Other mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, their representation theories; as outlined above, projective space is a geometric object that formalizes statements like "Parallel lines intersect at infinity." For concreteness, we give the construction of the real projective plane P2 in some detail. There are three equivalent definitions: The set of all lines in R3 passing through the origin.
Every such line meets the sphere of radius one centered in the origin twice, say in P = and its antipodal point. P2 can be described as the points on the sphere S2, where every point P and its antipodal point are not distinguished. For example, the point is identified with, etc, yet another equivalent definition is the set of equivalence classes of R3 ∖, i.e. 3-space without the origin, where two points P = and P∗ = are equivalent iff there is a nonzero real number λ such that P = λ⋅P∗, i.e. x = λx∗, y = λy∗, z = λz∗. The usual way to write an element of the projective plane, i.e. the equivalence class corresponding to an honest point in R3, is. The last formula goes under the name of homogeneous coordinates. In homogeneous coordinates, any point with z ≠ 0 is equivalent to. So there are two disjoint subsets of the projective plane: that consisting of the points = for z ≠ 0, that consisting of the remaining points; the latter set can be subdivided into two disjoint subsets, with points and. In the last case, x is nonzero, because the origin was not part of P2.
This last point is equivalent to. Geometrically, the first subset, isomorphic to R2, is in the image the yellow upper hemisphere, or equivalently the lower hemisphere; the second subset, isomorphic to R1, corresponds to the green line, or, equivalently the light green line. We have the red point or the equivalent light red point. We thus have a disjoint decomposition P2 = R2 ⊔ R1 ⊔ point. Intuitively, made precise below, R1 ⊔ point is itself the real projective line P1. Considered as a subset of P2, it is called line at infinity, whereas R2 ⊂ P2 is called affine plane, i.e. just the usual plane. The next objective is to make the saying "parallel lines meet at infinity" precise. A natural bijection between the plane z = 1 and the sphere of the projective plane is accomplished by the gnomonic projection; each point P on this plane is mapped to the two intersection points of the sphere with the line through its center and P. These two points are identified in the projective plane. Lines in the plane are mapped to great circles if one includes one pair of antipodal points on the equator.
Any two great circles intersect in two antipodal points. Great circles corresponding to parallel lines intersect on the equator. So any two lines have one intersection point inside P2; this phenomenon is axiomatized in projective geometry. The real projective space of dimension n or projective n-space, Pn, is the set of the lines in Rn+1 passing through the origin. For defining it as a topological space and as an algebraic variety it is better to define it as the quotient space of Rn+1 by the equivalence relation "to be aligned with the origin". More Pn:= / ~,where ~ is the equivalence relation defined by: ~ if there is a non-zero real number λ such that =; the elements of the projective space are called points. The projective coordinates of a point P are x0... xn, where is any element of the corresponding equivalence class. This is denoted P =, the colons and the brackets emphasizing that the right-hand side is an equivalence class, whic