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David Feldman
David Feldman
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The totality of all holomorphic functions on the unit disk forms some sort of infinite-dimensional complex manifold, where the coefficients of the TalyorTaylor expansion might serve as coordinates for the space.

Passing from the functions to their zero-sets fibers this set over the space of infinite discrete subsets of the disk, also some sort of infinite-dimensional complex manifold.

I'd like to know if there's a general obstruction to the existence of any kind of holomorphic cross-section. Perhaps even there are no non-constant holomorphic functions on the base space at all?

I understand that there's no naive generalization available from the usual elementary symmetric polynomials to infinitely many variables. Indeed the delicate nature of the Weierstrass factorization theorem reflects that. So I suppose I'm looking for a general principle that would imply that any Weierstrass-like theorem must involve, in an essential way, putting something like an order on the zero set, so that the output can't vary nicely across all zero-sets.

The totality of all holomorphic functions on the unit disk forms some sort infinite-dimensional complex manifold, where the coefficients of the Talyor expansion might serve as coordinates for the space.

Passing from the functions to their zero-sets fibers this set over the space of infinite discrete subsets of the disk, also some sort of infinite-dimensional complex manifold.

I'd like to know if there's a general obstruction to the existence of any kind of holomorphic cross-section. Perhaps even there are no non-constant holomorphic functions on the base space at all?

I understand that there's no naive generalization available from the usual elementary symmetric polynomials to infinitely many variables. Indeed the delicate nature of the Weierstrass factorization theorem reflects that. So I suppose I'm looking for a general principle that would imply that any Weierstrass-like theorem must involve, in an essential way, putting something like an order on the zero set, so that the output can't vary nicely across all zero-sets.

The totality of all holomorphic functions on the unit disk forms some sort of infinite-dimensional complex manifold, where the coefficients of the Taylor expansion might serve as coordinates for the space.

Passing from the functions to their zero-sets fibers this set over the space of infinite discrete subsets of the disk, also some sort of infinite-dimensional complex manifold.

I'd like to know if there's a general obstruction to the existence of any kind of holomorphic cross-section. Perhaps even there are no non-constant holomorphic functions on the base space at all?

I understand that there's no naive generalization available from the usual elementary symmetric polynomials to infinitely many variables. Indeed the delicate nature of the Weierstrass factorization theorem reflects that. So I suppose I'm looking for a general principle that would imply that any Weierstrass-like theorem must involve, in an essential way, putting something like an order on the zero set, so that the output can't vary nicely across all zero-sets.

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David Feldman
David Feldman
  • 17.6k
  • 8
  • 67
  • 135

Basic obstruction to anything like holomophic symmetric functions of infinitely many variables?

The totality of all holomorphic functions on the unit disk forms some sort infinite-dimensional complex manifold, where the coefficients of the Talyor expansion might serve as coordinates for the space.

Passing from the functions to their zero-sets fibers this set over the space of infinite discrete subsets of the disk, also some sort of infinite-dimensional complex manifold.

I'd like to know if there's a general obstruction to the existence of any kind of holomorphic cross-section. Perhaps even there are no non-constant holomorphic functions on the base space at all?

I understand that there's no naive generalization available from the usual elementary symmetric polynomials to infinitely many variables. Indeed the delicate nature of the Weierstrass factorization theorem reflects that. So I suppose I'm looking for a general principle that would imply that any Weierstrass-like theorem must involve, in an essential way, putting something like an order on the zero set, so that the output can't vary nicely across all zero-sets.