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A map of IndRings is a map of Ind-objects in finitely generated k-algebras. That is, it is given by specifying countable subsets S and T of the natural numbers and a surjective lax order preserving map f from S to T and a map of k-algebras from the k-algebra associated to an element s in S to the k-algebra associated to f(s). These maps are supposed to commute with all of the structure, but we will not check this in the software.
We also provide a way to promote a finitely generated k-algebra to an IndRing.
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+Given an IndRing, we define a n-variable power series over it by specifying a function 'b' from the natural numbers to the natural numbers and then a function from the n-tuples of natural numbers of total degree i (just the sum of the values in the n-tuple) to the b(i)'th ring in the IndRing. When the user asks to view the power series to degree 'i', the result will be a polynomial of total degree i over the b(i)'th ring in the IndRing.
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The goal of this project is to build software in Sage for doing lazy computation with multivariable formal power series over large commutative ring (such as the Lazard ring, which is a polynomial ring over the integers in an countable number of variables).
We will define power series by describing a function from a cartesian power of the natural numbers to the coefficient ring. We plan to manipulate such power series in a completely formal manner. For example, when two formal power series are added, the resulting formal power series is just a tree containing the node "+" with the two functions defining the formal power series as children. The only time computation of the coefficients occurs is when the user asks to "view" a power series out to some range. Then our goal is to perform the minimal amount of computation in order to show the resulting power series to the user to the specified precision.
We also introduce a class of commutative rings called "IndRings". They are simply sequential diagrams of finitely generated k-algebras, where k is a base ring that Sage knows about. An IndRing is defined by specifying a function from the natural numbers to finitely generated k-algebras and, for each natural number i, a map of k-algebras from the k-algebra associated to i to the k-algebra associated to i+1.
A map of IndRings is a map of Ind-objects in finitely generated k-algebras. That is, it is given by specifying countable subsets S and T of the natural numbers and a surjective lax order preserving map f from S to T and a map of k-algebras from the k-algebra associated to an element s in S to the k-algebra associated to f(s). These maps are supposed to commute with all of the structure, but we will not check this in the software.
We also provide a way to promote a finitely generated k-algebra to an IndRing.
Given an IndRing, we define a n-variable power series over it by specifying a function 'b' from the natural numbers to the natural numbers and then a function from the n-tuples of natural numbers of total degree i (just the sum of the values in the n-tuple) to the b(i)'th ring in the IndRing. When the user asks to view the power series to degree 'i', the result will be a polynomial of total degree i over the b(i)'th ring in the IndRing.
Relation to Power Series and Multivariate Power Series (the code already in Sage):
We built this code on top of the existing code. This code is not meant to be a replacement but to be an extension of the existing code.
As mentioned above, the formal power series in this code are functions from a cartesian power of the natural numbers to a ring (or IndRing as described above). These formal power series are manipulated completely formally, we keep track of a tree of operations that user has performed on their formal power series.
However, when the user asks to view one of these formal power series out to some degree, then the formal power series in the leaves of the tree are used to construct multivariate power series (in the sense of Sage) and these multivariate power series are manipulated according to the operations in the tree using the existing Sage code.
There are two points at which something strange happens.
First, if the user asks to apply reversion to a single variable formal power series, then we convert the multivariate power series into a power series (in the sense of Sage) and apply reversion and then convert the result back to a multivariable power series.
Second, we found the multiplication of multivariable power series to be slow. After looking at the code, we believe that this is due to the fact that the current code multiplies the underlying polynomials and then truncates the result. Of course, it is more efficient to only multiply terms that contribute to the product up to the degree that you are interested in. We have implemented this more efficient multiplication in this code and it is the default multiplication for formal power series. We also make use of this faster multiplication when the user asks to view the composite of certain formal power series.
CC: @nilesjohnson
Component: algebra
Keywords: formal power series, LazyPowerSeries
Author: Owen McGrath, Rick McQueen, Ethan Reed, Nathaniel Stapleton
Branch/Commit: u/gh-theowen/formal_power_series_over_large_commutative_rings @
d3f3434Issue created by migration from https://trac.sagemath.org/ticket/26549