Noether's advisor, Paul Gordan, was known as the " king of invariant theory ", and his chief contribution to mathematics was his 1870 solution of the finite basis problem for invariants of homogeneous polynomials in two variables.
42.
To illustrate this procedure, consider a group of-dimensional matrices as a subset of Euclidean space, and let the space of functions be polynomials, perhaps of some maximum degree, or even homogeneous polynomials of, all defined on.
43.
The difference between the " k " th and ( k-1 ) st functors is a " homogeneous functor of degree " k " " ( by analogy with homogeneous polynomials ), which can be classified.
44.
One can see further that the space of homogeneous polynomials of degree " k " can be identified with the symmetric tensor power S ^ k \ mathbb { C } ^ n of the standard representation \ mathbb C ^ n.
45.
Comparing this to the definition of " O " ( 1 ), above, we see that the sections of " O " ( 1 ) are in fact linear homogeneous polynomials, generated by the x _ i themselves.
46.
Note that every closed subscheme of projective space can be defined as the zero set of some collection of homogeneous polynomials, hence as the zero set of some sections of the line bundles " O " ( " j " ).
47.
:If no seven points out of lie on a non-degenerate conic, and no four points out of lie on a line, then the vector space of cubic homogeneous polynomials that vanish on ( the affine cones of ) has dimension two.
48.
Applying this to the sentence stating that every non-constant homogeneous polynomial of degree " d " in at least " d " 2 + 1 variables represents 0, and using Lang's theorem, one gets the Ax Kochen theorem.
49.
This allows one to define holomorphic and antiholomorphic one-forms and ( " m, n " )-forms, which are homogeneous polynomials in these one-forms with " m " holomorphic factors and " n " antiholomorphic factors.
50.
The determinant of the modified Laplacian matrix by deleting any row and column ( similar to finding the number of spanning trees from the original Laplacian matrix ), above is then a homogeneous polynomial ( the Kirchhoff polynomial ) in the indeterminants corresponding to the edges of the graph.
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