In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial in three variables.
Hilbert ( 1888 ) showed that a positive semi-definite ternary quartic form over the reals can be written as a sum of three squares of quadratic forms.
The ring of invariants is generated by 7 algebraically independent invariants of degrees 3, 6, 9, 12, 15, 18, 27 (discriminant) ( Dixmier 1987 ), together with 6 more invariants of degrees 9, 12, 15, 18, 21, 21, as conjectured by Shioda (1967). Salmon (1879) discussed the invariants of order up to about 15.
The Salmon invariant is a degree 60 invariant vanishing on ternary quartics with an inflection bitangent. ( Dolgachev 2012 , 6.4)
The catalecticant of a ternary quartic is the resultant of its 6 second partial derivatives. It vanishes when the ternary quartic can be written as a sum of five 4th powers of linear forms.
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. For example, if we consider the action of the special linear group SLn on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SLn.
In mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley (1846), Siegfried Heinrich Aronhold (1858), Alfred Clebsch (1861), and Paul Gordan (1887) in the 19th century for computing invariants of algebraic forms. It is based on treating the form as if it were a power of a degree one form, which corresponds to embedding a symmetric power of a vector space into the symmetric elements of a tensor product of copies of it.
Jean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of mathematics, particularly in the fields of functional analysis and algebraic topology. His work on the classical groups, and on formal groups, introducing what now are called Dieudonné modules, had a major effect on those fields.
In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form . The Kronecker–Weber theorem provides a partial converse: every finite abelian extension of Q is contained within some cyclotomic field. In other words, every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients. For example,
In mathematics, Harish-Chandra's regularity theorem, introduced by Harish-Chandra (1963), states that every invariant eigendistribution on a semisimple Lie group, and in particular every character of an irreducible unitary representation on a Hilbert space, is given by a locally integrable function. Harish-Chandra proved a similar theorem for semisimple p-adic groups.
In mathematics, a Kodaira surface is a compact complex surface of Kodaira dimension 0 and odd first Betti number. The concept is named after Kunihiko Kodaira.
In algebraic geometry, the Igusa quartic is a quartic hypersurface in 4-dimensional projective space, studied by Igusa (1962). It is closely related to the moduli space of genus 2 curves with level 2 structure. It is the dual of the Segre cubic.
In algebraic geometry, a Coble surface was defined by Dolgachev & Zhang (2001) to be a smooth rational projective surface with empty anti-canonical linear system |−K| and non-empty anti-bicanonical linear system |−2K|. An example of a Coble surface is the blowing up of the projective plane at the 10 nodes of a Coble curve.
The Classical Groups: Their Invariants and Representations is a mathematics book by Hermann Weyl (1939), which describes classical invariant theory in terms of representation theory. It is largely responsible for the revival of interest in invariant theory, which had been almost killed off by David Hilbert's solution of its main problems in the 1890s.
In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary form in two variables x and y that remains invariant under the special linear group acting on the variables x and y.
In mathematical invariant theory, the catalecticant of a form of even degree is a polynomial in its coefficients that vanishes when the form is a sum of an unusually small number of powers of linear forms. It was introduced by Sylvester (1852); see Miller (2010). The word catalectic refers to an incomplete line of verse, lacking a syllable at the end or ending with an incomplete foot.
In the mathematical theory of Kleinian groups, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem was proved by Lars Ahlfors, apart from a gap that was filled by Greenberg (1967).
In the mathematical field of convex geometry, the Busemann–Petty problem, introduced by Herbert Busemann and Clinton Myers Petty, asks whether it is true that a symmetric convex body with larger central hyperplane sections has larger volume. More precisely, if K, T are symmetric convex bodies in Rn such that
In mathematics, a nullform of a vector space acted on linearly by a group is a vector on which all invariants of the group vanish. Nullforms were introduced by Hilbert (1893)..
The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David Hilbert and the Italian school of algebraic geometry in the beginning of the century, and later formalized by André Weil, Jean-Pierre Serre and Alexander Grothendieck. Much of the classical terminology, mainly based on case study, was simply abandoned, with the result that books and papers written before this time can be hard to read. This article lists some of this classical terminology, and describes some of the changes in conventions.
This page is a glossary of terms in invariant theory. For descriptions of particular invariant rings, see invariants of a binary form, symmetric polynomials. For geometric terms used in invariant theory see the glossary of classical algebraic geometry. Definitions of many terms used in invariant theory can be found in, ,, ,, ,, , and the index to the fourth volume of Sylvester's collected works includes many of the terms invented by him.
In mathematical invariant theory, a perpetuant is informally an irreducible covariant of a form or infinite degree. More precisely, the dimension of the space of irreducible covariants of given degree and weight for a binary form stabilizes provided the degree of the form is larger than the weight of the covariant, and the elements of this space are called perpetuants. Perpetuants were introduced and named by Sylvester (1882, p.105). MacMahon and Stroh (1890) classified the perpetuants. Elliott (1907) describes the early history of perpetuants and gives an annotated bibliography.
In mathematics, a ternary cubic form is a homogeneous degree 3 polynomial in three variables.
In mathematics, a quaternary cubic form is a degree 3 homogeneous polynomial in four variables. The zeros form a cubic surface in 3-dimensional projective space.
In algebraic number theory, the Ferrero–Washington theorem, proved first by Ferrero & Washington (1979) and later by Sinnott (1984), states that Iwasawa's μ-invariant vanishes for cyclotomic Zp-extensions of abelian algebraic number fields.