# Multiplicative summations into algebraically closed fields

@inproceedings{Dawson2021MultiplicativeSI, title={Multiplicative summations into algebraically closed fields}, author={Robert J. MacG. Dawson and Grant Molnar}, year={2021} }

In this paper, extending our earlier program, we derive maximal canonical extensions for multiplicative summations into algebraically closed fields. We show that there is a well-defined analogue to minimal polynomials for a series algebraic over a ring of series, the “scalar polynomial”. When that ring is the domain of a summation S, we derive the related concepts of the S-minimal polynomial for a series, which is mapped by S to a scalar polynomial. When the scalar polynomial for a series has…

## References

SHOWING 1-10 OF 11 REFERENCES

Telescopic, Multiplicative, and Rational Extensions of Summations

- Mathematics
- 2021

A summation is a shift-invariant R-module homomorphism from a submodule of R[[σ]] to R or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this…

Ramanujan Summation of Divergent Series

- Mathematics
- 2017

In Chapter VI of his second Notebook Ramanujan introduce the Euler-MacLaurin formula to define the " constant " of a series. When the series is divergent he uses this " constant " like a sum of the…

Formal Summation of Divergent Series

- Physics
- 1998

The idea of telescoping a series is widely known, but is not widely trusted. It is often treated as a formalism with no meaning, unless convergence is already established. It is shown here that even…

An Introduction to Abstract Algebra

- Mathematics, Computer Science
- 2005

This chapter describes a variety of basic algebraic structures that play roles in the generation and analysis of sequences, especially sequences intended for use in communications and cryptography.

Zeta function regularization of path integrals in curved spacetime

- Mathematics
- 1977

This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator…

Divergent series, Éditions Jacques Gabay, Sceaux, 1992, With a preface by J. E. Littlewood and a note by L

- Reprint of the revised (1963) edition
- 1963

Adding, multiplying and the Mellin transform, August 2007, https://cornellmath.wordpress.com/2007/08/03/adding-multiplying-and-the-mellin-transform

- 2007

MacG . Dawson , Formal summation of divergent series

- J . Math . Anal . Appl .
- 1985

Ramanujan’s notebooks: Part 1, Springer-Verlag

- 1985