MilnorAnalyzer Class Reference

#include <derive.H>

Inheritance diagram for MilnorAnalyzer:

List of all members.

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Detailed Description

a class to compute the kernel of all Milnor derivations

Given an unstable algebra A, we are interested in computing the intersection of all the Milnor derivations Q_i, i=0, 1, 2... . When A is finitely generated this intersection "stabilizes" and thus can be computed in finite time. In fact, what we are after is the even part of this intersection.

In principle, one could use DerivationAnalyzer to compute the kernel of d= Q_0 = Sq^1, and then work on the subalgebra of constants, this time with d= Q_1, so as to obtain ker d = ker Q_0 inter ker Q_1, and so on and so forth. Indeed, in a sense this is exactly what we do.

However, in pratice we can avoid a lot of computations with some extra tweaking. Mostly, this means the following: say A is generated by x, y, and z, and ker Sq^1 is generated by sx, sy, sz and a_1, with sx= x^2 when viewed as an element in A, etc. Then, while studying Q_1, it would be wasteful to consider the squares ssx (=x^4...) ssy and ssz. So we prefer to keep sx, sy and sz in the algebra of squares, even though they are not squares in 'constants' (and from now on the term 'square' should be thought of as meaning 'a square somewhere', or more generally 'something that will surely be in the kernel of the derivations to consider subsequently'). So we end up implementing the improvements suggested in the comments to DerivationAnalyzer, and sx is kept in the algebra of squares because Q_1(sx)=0. Bottom line is, sx, sy and sz are in squares rather than ssx, ssy and ssz, and perhaps even more importantly, we can choose I (one) and a_1 as generators for 'constants' as a squares-modules -- no need for sxsy, etc.

Secondly, we note that a_1^2 is a square in A and thus can be written as a polynomial in sx, sy and sz. Therefore, while we thought that the new algebra of squares was a priori generated by sx, sy, sz and sa_1, this second remark now tells us that sa_1 is not useful, either. This will prevent 'squares' from growing embarrassingly large (however if Q_1(a_1)=0, we will add a_1 to squares).

So in the end we have been led to:

• write a method re_setup(), which will be called before computing the kernel of each Q_i, taking the previously obtained algebra of constants as the source, and setting everything up as we have discussed.
• write a method setup_for_Sq1(). The computation of Sq^1 is an exception, for 'squares' must initially be taken large enough to turn *source into a finitely generated module. This method is reasonably close to the original setup() of DerivationAnalyzer, but it does add variables with Sq^1=0 to squares.
• rewrite coords_of(), so as to replace, in the example above, an occurence of a_i^2 by its expression in terms of sx, sy and sz. This relies on information kept in newvar_squared (which in turn is filled by re_setup()). Again in the case of Sq^1, things are quite different, and in fact newvar_squared in this case is used to emulate the old behaviour of coords_of (that is, its behaviour in DerivationAnalyzer).
• Finally, there is start() which runs all the computations and checks for completeness (calling done() for this).

Important: A must have a Grobner basis.

Definition at line 237 of file derive.H.

Public Member Functions
bool	done ()
	checks whether we are done, and compute the final answer
virtual multiPolynomial< F_2 >	coords_of (PowProd P)
virtual multiPolynomial< F_2 >	coords_of (const Polynomial< F_2 > &P)
	given a member of source, this computes its coordinates when source is viewed as a squares-module
void	start (UnstableAlgebra *A)
	computes the kernel of Milnor derivations until their intersection is Steenrod stable
void	setup_for_Sq1 ()
	gets ready for the 1st computation
void	re_setup (long k)
	gets ready for the next computation
Public Attributes
UnstableAlgebra *	A
	the unstable algebra to analyze
GradedAlgebra< F_2 >	ans
	the answer
AffineHomomorphism< F_2 >	inclusion
	the inclusion of the constants into *A
AffineHomomorphism< F_2 >	ans_inclusion
	inclusion of the answer into *A
AbstractHomomorphism< F_2 >	newvar_squared
	expressing the squares of new variables in terms of standard squares
long	n_cs_vars
	an index to keep track of variables in the kernel of d
GradedAlgebra< F_2 >	alg
	a place to store 'constants' as the new *source

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Member Function Documentation

multiPolynomial< F_2 > MilnorAnalyzer::coords_of (  const Polynomial< F_2 > &  P  )  [virtual]

given a member of source, this computes its coordinates when source is viewed as a squares-module Reimplemented from DerivationAnalyzer. Definition at line 615 of file derive.cpp. References DerivationAnalyzer::coords_of(). Referenced by re_setup(), and setup_for_Sq1().

bool MilnorAnalyzer::done (   )

checks whether we are done, and compute the final answer This contructs the even part of 'constants', and checks whether its image in *A is Steenrod stable. If so, we are done. Definition at line 620 of file derive.cpp. References A, ans_inclusion, GradedAlgebra< K >::even_subalgebra(), UnstableAlgebra::has_closure_of(), inclusion, and AffineHomomorphism< K >::populate_associated_algebra(). Referenced by start().

void MilnorAnalyzer::re_setup (  long  k  )

gets ready for the next computation Swaps constants and *source, populates newvar_squared, computes the induced derivation on *source. Definition at line 444 of file derive.cpp. References A, alg, Polynomial< K >::alphabet, AbstractHomomorphism< K >::clear(), coords_of(), AffineHomomorphism< K >::fix_alphabets(), AffineAlgebra< K >::fix_alphabets(), AffineAlgebra< K >::get_variables(), GradedAlgebra< K >::hom_degrees, inclusion, Polynomial< K >::is_zero(), AffineHomomorphism< K >::maps_onto(), n_cs_vars, SimpleAlphabet::nameof(), GradedAlgebra< K >::new_variable(), newvar_squared, AbstractHomomorphism< K >::of(), AbstractHomomorphism< K >::of_generator(), AffineAlgebra< K >::one(), AffineHomomorphism< K >::populate_associated_algebra(), UnstableAlgebra::Q(), AffineAlgebra< K >::reduced_form(), Tuple< T >::resize(), AbstractHomomorphism< K >::set_image(), Tuple< T >::size(), AffineHomomorphism< K >::source, AbstractHomomorphism< K >::swap(), GradedAlgebra< K >::swap_variables_order(), AffineHomomorphism< K >::target, SimpleAlphabet::variables_in_use(), and DerivationAnalyzer::write_gens(). Referenced by start().

void MilnorAnalyzer::setup_for_Sq1 (   )

gets ready for the 1st computation Replaces setup(). Definition at line 327 of file derive.cpp. References A, alg, AbstractHomomorphism< K >::clear(), coords_of(), AffineAlgebra< K >::fix_alphabets(), AffineAlgebra< K >::get_variables(), GradedAlgebra< K >::hom_degrees, inclusion, Polynomial< K >::is_zero(), AffineHomomorphism< K >::maps_onto(), n_cs_vars, SimpleAlphabet::nameof(), GradedAlgebra< K >::new_variable(), newvar_squared, AbstractHomomorphism< K >::of(), AffineAlgebra< K >::one(), AffineHomomorphism< K >::populate_associated_algebra(), AffineAlgebra< K >::reduced_form(), AffineAlgebra< K >::relations, Tuple< T >::resize(), AbstractHomomorphism< K >::set_image(), Tuple< T >::size(), AffineHomomorphism< K >::source, UnstableAlgebra::Sq(), AbstractHomomorphism< K >::swap(), GradedAlgebra< K >::swap_variables_order(), AffineHomomorphism< K >::target, SimpleAlphabet::variables_in_use(), and DerivationAnalyzer::write_gens(). Referenced by start().

void MilnorAnalyzer::start (  UnstableAlgebra *  A  )

computes the kernel of Milnor derivations until their intersection is Steenrod stable Definition at line 276 of file derive.cpp. References A, DerivationAnalyzer::constant_subalgebra(), done(), inclusion, AffineHomomorphism< K >::populate_associated_algebra(), re_setup(), and setup_for_Sq1().

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Member Data Documentation

UnstableAlgebra* MilnorAnalyzer::A

the unstable algebra to analyze Definition at line 240 of file derive.H. Referenced by done(), re_setup(), setup_for_Sq1(), and start().

GradedAlgebra<F_2> MilnorAnalyzer::alg

a place to store 'constants' as the new *source Definition at line 292 of file derive.H. Referenced by re_setup(), and setup_for_Sq1().

GradedAlgebra<F_2> MilnorAnalyzer::ans

the answer Definition at line 242 of file derive.H.

AffineHomomorphism<F_2> MilnorAnalyzer::ans_inclusion

inclusion of the answer into *A When the algorithm terminates, 'constants' is the intersection of the kernels of some Milnor derivations, and 'ans' is the even part of that (which is comprised of elements belonging to ALL the kernels). This is the inclusion of the even part into *A. Definition at line 261 of file derive.H. Referenced by done().

AffineHomomorphism<F_2> MilnorAnalyzer::inclusion

the inclusion of the constants into *A While cs_inlusion is the inclusion of 'constants' into *source (the temporary algebra under analysis), this homomorphism is the inclusion of 'constants' all the way up to *A. Definition at line 250 of file derive.H. Referenced by done(), re_setup(), setup_for_Sq1(), and start().

long MilnorAnalyzer::n_cs_vars

an index to keep track of variables in the kernel of d n_cs_vars stands for "number of constants variables". Here constant means "constant a priori" -- which in most of the comments we have called a "square" (sorry for the slight inconsistency). Most of the time, n_cs_vars is equal to squares.variables_in_use(), but not quite always. If you look at the code you will see its use, otherwise you never have to worry about this. Definition at line 290 of file derive.H. Referenced by re_setup(), and setup_for_Sq1().

AbstractHomomorphism<F_2> MilnorAnalyzer::newvar_squared

expressing the squares of new variables in terms of standard squares After at least a first run, constants is obtained from squares by adding some variables, say a_1 ... a_n. Then a_i^2 is in squares. This AbstractHomomorphism will hold an expression for a_i^2 given by newvar_squared.of_generator( i + (nb of variables in squares)). Definition at line 279 of file derive.H. Referenced by re_setup(), and setup_for_Sq1().

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The documentation for this class was generated from the following files:

• include/derive.H
• src/derive.cpp

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