Algebraic invariant theory Emmy Noether

table 2 noether s dissertation on invariant theory. table collects 202 of 331 invariants of ternary biquadratic forms. these forms graded in 2 variables x , u. horizontal direction of table lists invariants increasing grades in x, while vertical direction lists them increasing grades in u.

much of noether s work in first epoch of career associated invariant theory, principally algebraic invariant theory. invariant theory concerned expressions remain constant (invariant) under group of transformations. everyday example, if rigid yardstick rotated, coordinates (x1, y1, z1) , (x2, y2, z2) of endpoints change, length l given formula l = Δx + Δy + Δz remains same. invariant theory active area of research in later nineteenth century, prompted in part felix klein s erlangen program, according different types of geometry should characterized invariants under transformations, e.g., cross-ratio of projective geometry. archetypal example of invariant discriminant b − 4ac of binary quadratic form ax + bxy + cy. called invariant because unchanged linear substitutions x→ax + by, y→cx + dy determinant ad − bc = 1. these substitutions form special linear group sl2. (there no invariants under general linear group of invertible linear transformations because these transformations can multiplication scaling factor. remedy this, classical invariant theory considered relative invariants, forms invariant scale factor.) 1 can ask polynomials in a, b, , c unchanged action of sl2; these called invariants of binary quadratic forms, , turn out polynomials in discriminant. more generally, 1 can ask invariants of homogeneous polynomials a0xy + ... + arxy of higher degree, polynomials in coefficients a0, ..., ar, , more still, 1 can ask similar question homogeneous polynomials in more 2 variables.

one of main goals of invariant theory solve finite basis problem . sum or product of 2 invariants invariant, , finite basis problem asked whether possible invariants starting finite list of invariants, called generators, , then, adding or multiplying generators together. example, discriminant gives finite basis (with 1 element) invariants of binary quadratic forms. noether s advisor, paul gordan, known king of invariant theory , , chief contribution mathematics 1870 solution of finite basis problem invariants of homogeneous polynomials in 2 variables. proved giving constructive method finding of invariants , generators, not able carry out constructive approach invariants in 3 or more variables. in 1890, david hilbert proved similar statement invariants of homogeneous polynomials in number of variables. furthermore, method worked, not special linear group, of subgroups such special orthogonal group. first proof caused controversy because did not give method constructing generators, although in later work made method constructive. thesis, noether extended gordan s computational proof homogeneous polynomials in 3 variables. noether s constructive approach made possible study relationships among invariants. later, after had turned more abstract methods, noether called thesis mist (crap) , formelngestrüpp (a jungle of equations).