Tabulation of Cubic Function Fields Via Polynomial Binary Cubic Forms
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On the Number of Cubic Orders of Bounded Discriminant Having
On the number of cubic orders of bounded discriminant having automorphism group C3, and related problems Manjul Bhargava and Ariel Shnidman August 8, 2013 Abstract For a binary quadratic form Q, we consider the action of SOQ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegenerate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed “lattice shape”. Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with the discriminant of the corresponding cubic order. We use this interpretation of the integral orbits to solve three classical-style count- ing problems related to cubic orders and fields. First, we give an asymptotic formula for the number of cubic orders having bounded discriminant and nontrivial automor- phism group. More generally, we give an asymptotic formula for the number of cubic orders that have bounded discriminant and any given lattice shape (i.e., reduced trace form, up to scaling). Via a sieve, we also count cubic fields of bounded discriminant whose rings of integers have a given lattice shape. We find, in particular, that among cubic orders (resp. fields) having lattice shape of given discriminant D, the shape is equidistributed in the class group ClD of binary quadratic forms of discriminant D. As a by-product, we also obtain an asymptotic formula for the number of cubic fields of bounded discriminant having any given quadratic resolvent field. -
Math 2250 HW #10 Solutions
Math 2250 Written HW #10 Solutions 1. Find the absolute maximum and minimum values of the function g(x) = e−x2 subject to the constraint −2 ≤ x ≤ 1. Answer: First, we check for critical points of g, which is differentiable everywhere. By the Chain Rule, 2 2 g0(x) = e−x · (−2x) = −2xe−x : Since e−x2 > 0 for all x, g0(x) = 0 when 2x = 0, meaning when x = 0. Hence, x = 0 is the only critical point of g. Now we just evaluate g at the critical point and the endpoints: 2 g(−2) = e−(−2) = e−4 = 1=e4 2 g(0) = e−0 = e0 = 1 2 g(2) = e−1 = e−1 = 1=e: Since 1 > 1=e > 1=e4, we see that the absolute maximum of g(x) on this interval is at (0; 1) and the absolute minimum is at (−2; 1=e4). 2. Find all local maxima and minima of the curve y = x2 ln x. Answer: Notice, first of all, that ln x is only defined for x > 0, so the function x2 ln x is likewise only defined for x > 0. This function is differentiable on its entire domain, so we differentiate in search of critical points. Using the Product Rule, 1 y0 = 2x ln x + x2 · = 2x ln x + x = x(2 ln x + 1): x Since we're only allowed to consider x > 0, we see that the derivative is zero only when 2 ln x + 1 = 0, meaning ln x = −1=2. Therefore, we have a critical point at x = e−1=2. -
On the Number of Cubic Orders of Bounded Discriminant Having Automorphism Group C3, and Related Problems Manjul Bhargava and Ariel Shnidman
Algebra & Number Theory Volume 8 2014 No. 1 On the number of cubic orders of bounded discriminant having automorphism group C3, and related problems Manjul Bhargava and Ariel Shnidman msp ALGEBRA AND NUMBER THEORY 8:1 (2014) msp dx.doi.org/10.2140/ant.2014.8.53 On the number of cubic orders of bounded discriminant having automorphism group C3, and related problems Manjul Bhargava and Ariel Shnidman For a binary quadratic form Q, we consider the action of SOQ on a 2-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegen- erate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed “lattice shape”. Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with the discriminant of the corresponding cubic order. We use this interpretation of the integral orbits to solve three classical-style counting problems related to cubic orders and fields. First, we give an asymp- totic formula for the number of cubic orders having bounded discriminant and nontrivial automorphism group. More generally, we give an asymptotic formula for the number of cubic orders that have bounded discriminant and any given lattice shape (i.e., reduced trace form, up to scaling). Via a sieve, we also count cubic fields of bounded discriminant whose rings of integers have a given lattice shape. We find, in particular, that among cubic orders (resp. -
On Cubic Hypersurfaces of Dimension Seven and Eight
ON CUBIC HYPERSURFACES OF DIMENSION SEVEN AND EIGHT ATANAS ILIEV AND LAURENT MANIVEL Abstract. Cubic sevenfolds are examples of Fano manifolds of Calabi- Yau type. We study them in relation with the Cartan cubic, the E6- invariant cubic in P26. We show that a generic cubic sevenfold X can be described as a linear section of the Cartan cubic, in finitely many ways. To each such “Cartan representation” we associate a rank nine vector bundle on X with very special cohomological properties. In particular it allows to define auto-equivalences of the non-commutative Calabi- Yau threefold associated to X by Kuznetsov. Finally we show that the generic eight dimensional section of the Cartan cubic is rational. 1. Introduction Manifolds of Calabi-Yau type were defined in [IMa2] as compact complex manifolds of odd dimension whose middle dimensional Hodge structure is similar to that of a Calabi-Yau threefold. Certain manifolds of Calabi-Yau type were used in order to construct mirrors of rigid Calabi-Yau threefolds, and in several respects they behave very much like Calabi-Yau threefolds. This applies in particular to the cubic sevenfold, which happens to be the mirror of the so-called Z-variety – a rigid Calabi-Yau threefold obtained as a desingularization of the quotient of a product of three Fermat cubic curves by the group Z3 × Z3, see e.g. [Sch], [CDP], [BBVW]. In some way, manifolds of Calabi-Yau type come just after manifolds of K3 type: manifolds of even dimension 2n, whose middle dimensional Hodge structure is similar to that of a K3 surface. -
Supplement 1: Toolkit Functions
Supplement 1: Toolkit Functions What is a Function? The natural world is full of relationships between quantities that change. When we see these relationships, it is natural for us to ask “If I know one quantity, can I then determine the other?” This establishes the idea of an input quantity, or independent variable, and a corresponding output quantity, or dependent variable. From this we get the notion of a functional relationship in which the output can be determined from the input. For some quantities, like height and age, there are certainly relationships between these quantities. Given a specific person and any age, it is easy enough to determine their height, but if we tried to reverse that relationship and determine age from a given height, that would be problematic, since most people maintain the same height for many years. Function Function: A rule for a relationship between an input, or independent, quantity and an output, or dependent, quantity in which each input value uniquely determines one output value. We say “the output is a function of the input.” Function Notation The notation output = f(input) defines a function named f. This would be read “output is f of input” Graphs as Functions Oftentimes a graph of a relationship can be used to define a function. By convention, graphs are typically created with the input quantity along the horizontal axis and the output quantity along the vertical. The most common graph has y on the vertical axis and x on the horizontal axis, and we say y is a function of x, or y = f(x) when the function is named f. -
Cubic Rings and Forms, II
CUBIC RINGS AND FORMS ILA VARMA The study of the link between integral forms and rings that are free of finite rank as Z-modules is generally motivated by the study and classification of such rings. Here, we see what the ring structure can tell us about representation of integers by certain forms, particularly in the case of cubics. This lecture is derived from a question posed by Manjul Bhargava at the Arizona Winter School on quadratic forms in March 2009. Definition 1. A ring of rank n is a commutative ring R with identity such that when viewed as a Z-module, it is free of rank n. The usual example for rings of rank n will be orders in number fields of degree n. However, note that these rings are a priori also integral domains, i.e. not the most generic of examples. In studying all rings of rank n, the basic question to ask is how one can classify them (up to ring isomorphism) for a fixed rank. As we will see, the answer for small n is very much related to integral binary forms. 1. n < 3 For n = 0, we have the zero ring consisting of 1 element, the additive and multiplicative identity. For n = 1, the free Z-module structure restricts us to having only one possibility, Z. The story becomes a bit more interesting for n = 2. To start as simply as possible, note that the information on the structure of a unique (up to isomorphism) ring of rank 2 is in its multiplication table. -
As1.4: Polynomial Long Division
AS1.4: POLYNOMIAL LONG DIVISION One polynomial may be divided by another of lower degree by long division (similar to arithmetic long division). Example (x32+ 3 xx ++ 9) (x + 2) x + 2 x3 + 3x2 + x + 9 1. Write the question in long division form. 3 2. Begin with the x term. 2 x3 divided by x equals x2. x 2 Place x above the division bracket x+2 x32 + 3 xx ++ 9 as shown. subtract xx32+ 2 2 3. Multiply x + 2 by x . x2 xx2+=+22 x 32 x ( ) . Place xx32+ 2 below xx32+ 3 then subtract. x322+−+32 x x xx = 2 ( ) 4. “Bring down” the next term (x) as 2 x + x indicated by the arrow. +32 +++ Repeat this process with the result until xxx23x 9 there are no terms remaining. x32+↓2x 2 2 5. x divided by x equals x. x + x Multiply xx( +=+22) x2 x . xx2 +−1 ( xx22+−) ( x +2 x) =− x 32 x+23 x + xx ++ 9 6. Bring down the 9. xx32+2 ↓↓ 2 7. − x divided by x equals − 1. xx+↓ Multiply xx2 +↓2 −12( xx +) =−− 2 . −+x 9 (−+xx9) −−−( 2) = 11 −−x 2 The remainder is 11 +11 (x32+ 3 xx ++ 9) equals xx2 +−1, with remainder 11. (x + 2) AS 1.4 – Polynomial Long Division Page 1 of 4 June 2012 If the polynomial has an ‘xn ‘ term missing, add the term with a coefficient of zero. Example (2xx3 − 31 +÷) ( x − 1) Rewrite (2xx3 −+ 31) as (2xxx32+ 0 −+ 31) Divide using the method from the previous example. 2 22xx32÷= x 2xx+− 21 2 32 −= − 2xx( 12) x 2 x 32 x−12031 xxx + −+ 20xx32−=−−(22xx32) 2x2 2xx32− 2 ↓↓ 2 22xx÷= x 2xx2 −↓ 3 2xx( −= 12) x2 − 2 x 2 2 2 2xx−↓ 2 23xx−=−−22xx− x ( ) −+x 1 −÷xx =−1 −+x 1 −11( xx −) =−+ 1 0 −+xx1− ( + 10) = Remainder is 0 (2xx32− 31 +÷) ( x −= 1) ( 2 xx + 21 −) with remainder = 0 ∴(2xx32 − 3 += 1) ( x − 12)( xx + 2 − 1) See Exercise 1. -
Automorphisms of Cubic Surfaces in Positive Characteristic
Izvestiya: Mathematics 83:3 Izvestiya RAN : Ser. Mat. 83:3 5–82 DOI: https://doi.org/10.1070/IM8803 Automorphisms of cubic surfaces in positive characteristic I. Dolgachev and A. Duncan Abstract. We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely character- ize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero. Keywords: cubic surfaces, automorphisms, positive characteristic. In remembrance of Vassily Iskovskikh § 1. Introduction Let X be a smooth cubic surface in P3 defined over an algebraically closed field k of arbitrary characteristic p. The primary purpose of this paper is to classify the possible automorphism groups of X. Along the way, we also classify the possible automorphism groups of del Pezzo surfaces of degree 4. This is progress towards the larger goal of classifying finite subgroups of the plane Cremona group in positive characteristic (see [1] for characteristic 0). In characteristic 0, the first attempts at a classification of cubic surfaces were undertaken by S. Kantor [2], then A. Wiman [3] in the late nineteenth century. In 1942, B. Segre [4] classified all automorphism groups in his book on cubic surfaces. Unfortunately, all of these classification had errors. -
ON BINARY CUBIC and QUARTIC FORMS 3 Group 2 2 Unless the Invariants I(F ), J(F ) Vanishes
ON BINARY CUBIC AND QUARTIC FORMS STANLEY YAO XIAO Abstract. In this paper we determine the group of rational automorphisms of binary cubic and quartic forms with integer coefficients and non-zero discriminant in terms of certain quadratic covari- ants of cubic and quartic forms. This allows one to give precise asymptotic formulae for the number of integers in an interval representable by a binary cubic or quartic form and extends work of Hooley. Further, we give the field of definition of lines contained in certain cubic and quartic surfaces related to binary cubic and quartic forms. 1. Introduction Let F be a binary form with integer coefficients, non-zero discriminant, and degree d 3. For ≥ each positive number Z, put RF (Z) for the number of integers of absolute value at most Z which is representable by the binary form F . In [14], [16], and [17] C. Hooley gave explicitly the asymptotic formula for the quantity RF (Z) when F is an irreducible binary cubic form or a biquadratic quartic form. Various authors have dealt with the case when F is a diagonal form; see [26] for a summary of these results. In [26], Stewart and Xiao proved the existence of an asymptotic formula for RF (Z) for all F with integer coefficients, non-zero discriminant, and degree d 3. More precisely, they proved that for ≥ each such F , there exists a positive rational number WF which depends only on F for which the asymptotic formula 2 (1.1) RF (Z) WF AF Z d ∼ holds with a power-saving error term. -
Arxiv:1912.07347V1 [Math.AG] 16 Dec 2019
TWENTY-SEVEN QUESTIONS ABOUT THE CUBIC SURFACE KRISTIAN RANESTAD - BERND STURMFELS We present a collection of research questions on cubic surfaces in 3-space. These questions inspired a collection of papers to be published in a special issue of the journal Le Matematiche. This article serves as the introduc- tion to that issue. The number of questions is meant to match the number of lines on a cubic surface. We end with a list of problems that are open. 1. Introduction One of most prominent results in classical algebraic geometry, derived two cen- turies ago by Cayley and Salmon, states that every smooth cubic surface in com- plex projective 3-space P3 contains 27 straight lines. This theorem has inspired generations of mathematicians, and it is still of importance today. The advance of tropical geometry and computational methods, and the strong current interest in applications of algebraic geometry, led us to revisit the cubic surface. Section 3 of this article gives a brief exposition of the history of that classical subject and how it developed. We offer a number of references that can serve as first entry points for students of algebraic geometry who wish to learn more. In December 2018, the second author compiled the list of 27 question. His original text was slightly edited and it now constitutes our Section 2 below. These questions were circulated, and they were studied by various groups of young mathematicians, in particular in Leipzig and Berlin. In May 2019, a one- day seminar on cubic surfaces was held in Oslo. Different teams worked on the arXiv:1912.07347v1 [math.AG] 16 Dec 2019 questions and they made excellent progress. -
Mathematics of the Gateway Arch Page 220
ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com Highlights in Springer’s eBook of the American Mathematical Society Collection February 2010 Volume 57, Number 2 An Invitation to Cauchy-Riemann NEW 4TH NEW NEW EDITION and Sub-Riemannian Geometries 2010. XIX, 294 p. 25 illus. 4th ed. 2010. VIII, 274 p. 250 2010. XII, 475 p. 79 illus., 76 in 2010. XII, 376 p. 8 illus. (Copernicus) Dustjacket illus., 6 in color. Hardcover color. (Undergraduate Texts in (Problem Books in Mathematics) page 208 ISBN 978-1-84882-538-3 ISBN 978-3-642-00855-9 Mathematics) Hardcover Hardcover $27.50 $49.95 ISBN 978-1-4419-1620-4 ISBN 978-0-387-87861-4 $69.95 $69.95 Mathematics of the Gateway Arch page 220 Model Theory and Complex Geometry 2ND page 230 JOURNAL JOURNAL EDITION NEW 2nd ed. 1993. Corr. 3rd printing 2010. XVIII, 326 p. 49 illus. ISSN 1139-1138 (print version) ISSN 0019-5588 (print version) St. Paul Meeting 2010. XVI, 528 p. (Springer Series (Universitext) Softcover ISSN 1988-2807 (electronic Journal No. 13226 in Computational Mathematics, ISBN 978-0-387-09638-4 version) page 315 Volume 8) Softcover $59.95 Journal No. 13163 ISBN 978-3-642-05163-0 Volume 57, Number 2, Pages 201–328, February 2010 $79.95 Albuquerque Meeting page 318 For access check with your librarian Easy Ways to Order for the Americas Write: Springer Order Department, PO Box 2485, Secaucus, NJ 07096-2485, USA Call: (toll free) 1-800-SPRINGER Fax: 1-201-348-4505 Email: [email protected] or for outside the Americas Write: Springer Customer Service Center GmbH, Haberstrasse 7, 69126 Heidelberg, Germany Call: +49 (0) 6221-345-4301 Fax : +49 (0) 6221-345-4229 Email: [email protected] Prices are subject to change without notice. -
Low-Degree Polynomial Roots
Low-Degree Polynomial Roots David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Created: July 15, 1999 Last Modified: September 10, 2019 Contents 1 Introduction 3 2 Discriminants 3 3 Preprocessing the Polynomials5 4 Quadratic Polynomials 6 4.1 A Floating-Point Implementation..................................6 4.2 A Mixed-Type Implementation...................................7 5 Cubic Polynomials 8 5.1 Real Roots of Multiplicity Larger Than One............................8 5.2 One Simple Real Root........................................9 5.3 Three Simple Real Roots......................................9 5.4 A Mixed-Type Implementation................................... 10 6 Quartic Polynomials 12 6.1 Processing the Root Zero...................................... 14 6.2 The Biquadratic Case........................................ 14 6.3 Multiplicity Vector (3; 1; 0; 0).................................... 15 6.4 Multiplicity Vector (2; 2; 0; 0).................................... 15 6.5 Multiplicity Vector (2; 1; 1; 0).................................... 15 6.6 Multiplicity Vector (1; 1; 1; 1).................................... 16 1 6.7 A Mixed-Type Implementation................................... 17 2 1 Introduction Consider a polynomial of degree d of the form d X i p(y) = piy (1) i=0 where the pi are real numbers and where pd 6= 0. A root of the polynomial is a number r, real or non-real (complex-valued with nonzero imaginary part) such that p(r) = 0. The polynomial can be factored as p(y) = (y − r)mf(y), where m is a positive integer and f(r) 6= 0.