Mixing
Intersection of parameterized cosets in a free-abelian group
$begingroup$
Let $L_1,L_2$ be subgroups of $mathbb{Z}^m$, and let $mathbf{P}_1,mathbf{P}_2$ be $rtimes m$ integer matrices. Then, it is straightforward to check that the set
begin{equation}
S_{{1,2}} := {winmathbb{Z}^r colon (wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) neq varnothing}
end{equation}
is the set (indeed subgroup) of preimages by $mathbf{P}_1 - mathbf{P}_2$ of the subgroup $L_1 + L_2$. That is,
begin{equation} label{eq: 2-intersection}
S_{{1,2}}
=
%{winmathbb{Z}^r colon (wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) neq varnothing}
%=
(L_1+L_2)(mathbf{P}_1 - mathbf{P}_2)^{-1}
leqslant
mathbb{Z}^r .
end{equation}
This concise description allows to use linear algebra to compute $S_{{1,2}}$, given
$L_1,L_2,mathbf{P}_1,mathbf{P}_2$.
Question:
I am interested in obtaining an analogous expression but with the intersection involving any finite (say $k$) number of cosets. More precisely, given $L_i$ subgroups of $mathbb{Z}^r$, and $mathbf{P}_i$ integer matrices of size $r times m$ $(iin I_k={1,ldots,k})$. I would like to obtain a "nice" description for the set
begin{equation}
S_{I_k}
=
left{
win mathbb{Z}^r :, bigcapnolimits_{i=1}^k (wmathbf{P}_i + L_i) neq varnothing
right},
end{equation}
in terms of the data $L_i,mathbf{P}_i$ that allows to compute it (or at least its index in $mathbb{Z}^r$).
Discussion of the case $k=3$.
Suppose that we want to describe the set:
begin{equation}
S_{{1,2,3}}
=
{
w in mathbb{Z}^r colon
(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) cap (wmathbf{P}_3 + L_3) neq varnothing
}.
end{equation}
Since the intersection of two cosets is either empty, or a coset of the intersection of the defining subgroups, one is tempted to
restrict the possible $w$'s in $S_{{1,2,3}}$ to $S_{{1,2}} = (L_1+L_2)(mathbf{P}_1 - mathbf{P}_2)^{-1}$ (since a triple intersection can only be nonempty if the double intersections are nonempty), write $(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2)$ as a coset of $L_1 cap L_2$ to reduce the triple intersection to a double intersection, and then reduce to the case $k=2$. Namely, write
begin{equation}
(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) cap (wmathbf{P}_3 + L_3)
=
(w' + (L_1 cap L_2) cap (wmathbf{P}_3 + L_3),
end{equation}
where w' is some element in the (now nonempty) intersection $(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2)$. The problem with this approach is that $w'$ depends on $w$ and so it does the exression we get for $S_{{1,2,3}}$, which, however, should depend only on $L_1,L_2,L_3,mathbf{P_1},mathbf{P_2},mathbf{P_3}$. Do you see a way (maybe another approach?) that allows to get rid of this dependence on w'?
Final remark.
Note that if the $L_i$'s are in direct sum (i.e., $sum_{iin I} L_i = bigoplus_{iin I} L_i$), then:
begin{equation}
bigcapnolimits_{i=1}^k (wmathbf{P}_i + L_i) neq varnothing
quad text{if and only if}quad
(wmathbf{P}_i + L_i) cap (wmathbf{P}_{i+1} + L_{i+1}) neq varnothing,
text{ for all } i text{ mod } k,
end{equation}
and hence, in this case $S_{I_k} = bigcap_{i text{ mod }k} S_{{i,i+1}}$, depending only in the initial data as we wanted.
Would it be possible to change the initial $L_i$'s by some $L'_i$'s in direct sum, such that the emptyness of the tested intersection does not change, and hence reduce the general case to the direct sum case?
linear-algebra matrices matrix-equations free-modules free-abelian-group
asked Jan 13 at 16:33
suitangisuitangi
45928
$endgroup$
add a comment |
$begingroup$
Let $L_1,L_2$ be subgroups of $mathbb{Z}^m$, and let $mathbf{P}_1,mathbf{P}_2$ be $rtimes m$ integer matrices. Then, it is straightforward to check that the set
begin{equation}
S_{{1,2}} := {winmathbb{Z}^r colon (wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) neq varnothing}
end{equation}
is the set (indeed subgroup) of preimages by $mathbf{P}_1 - mathbf{P}_2$ of the subgroup $L_1 + L_2$. That is,
begin{equation} label{eq: 2-intersection}
S_{{1,2}}
=
%{winmathbb{Z}^r colon (wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) neq varnothing}
%=
(L_1+L_2)(mathbf{P}_1 - mathbf{P}_2)^{-1}
leqslant
mathbb{Z}^r .
end{equation}
This concise description allows to use linear algebra to compute $S_{{1,2}}$, given
$L_1,L_2,mathbf{P}_1,mathbf{P}_2$.
Question:
I am interested in obtaining an analogous expression but with the intersection involving any finite (say $k$) number of cosets. More precisely, given $L_i$ subgroups of $mathbb{Z}^r$, and $mathbf{P}_i$ integer matrices of size $r times m$ $(iin I_k={1,ldots,k})$. I would like to obtain a "nice" description for the set
begin{equation}
S_{I_k}
=
left{
win mathbb{Z}^r :, bigcapnolimits_{i=1}^k (wmathbf{P}_i + L_i) neq varnothing
right},
end{equation}
in terms of the data $L_i,mathbf{P}_i$ that allows to compute it (or at least its index in $mathbb{Z}^r$).
Discussion of the case $k=3$.
Suppose that we want to describe the set:
begin{equation}
S_{{1,2,3}}
=
{
w in mathbb{Z}^r colon
(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) cap (wmathbf{P}_3 + L_3) neq varnothing
}.
end{equation}
Since the intersection of two cosets is either empty, or a coset of the intersection of the defining subgroups, one is tempted to
restrict the possible $w$'s in $S_{{1,2,3}}$ to $S_{{1,2}} = (L_1+L_2)(mathbf{P}_1 - mathbf{P}_2)^{-1}$ (since a triple intersection can only be nonempty if the double intersections are nonempty), write $(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2)$ as a coset of $L_1 cap L_2$ to reduce the triple intersection to a double intersection, and then reduce to the case $k=2$. Namely, write
begin{equation}
(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) cap (wmathbf{P}_3 + L_3)
=
(w' + (L_1 cap L_2) cap (wmathbf{P}_3 + L_3),
end{equation}
where w' is some element in the (now nonempty) intersection $(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2)$. The problem with this approach is that $w'$ depends on $w$ and so it does the exression we get for $S_{{1,2,3}}$, which, however, should depend only on $L_1,L_2,L_3,mathbf{P_1},mathbf{P_2},mathbf{P_3}$. Do you see a way (maybe another approach?) that allows to get rid of this dependence on w'?
Final remark.
Note that if the $L_i$'s are in direct sum (i.e., $sum_{iin I} L_i = bigoplus_{iin I} L_i$), then:
begin{equation}
bigcapnolimits_{i=1}^k (wmathbf{P}_i + L_i) neq varnothing
quad text{if and only if}quad
(wmathbf{P}_i + L_i) cap (wmathbf{P}_{i+1} + L_{i+1}) neq varnothing,
text{ for all } i text{ mod } k,
end{equation}
and hence, in this case $S_{I_k} = bigcap_{i text{ mod }k} S_{{i,i+1}}$, depending only in the initial data as we wanted.
Would it be possible to change the initial $L_i$'s by some $L'_i$'s in direct sum, such that the emptyness of the tested intersection does not change, and hence reduce the general case to the direct sum case?
linear-algebra matrices matrix-equations free-modules free-abelian-group
asked Jan 13 at 16:33
suitangisuitangi
45928
$endgroup$
add a comment |
$begingroup$
Let $L_1,L_2$ be subgroups of $mathbb{Z}^m$, and let $mathbf{P}_1,mathbf{P}_2$ be $rtimes m$ integer matrices. Then, it is straightforward to check that the set
begin{equation}
S_{{1,2}} := {winmathbb{Z}^r colon (wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) neq varnothing}
end{equation}
is the set (indeed subgroup) of preimages by $mathbf{P}_1 - mathbf{P}_2$ of the subgroup $L_1 + L_2$. That is,
begin{equation} label{eq: 2-intersection}
S_{{1,2}}
=
%{winmathbb{Z}^r colon (wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) neq varnothing}
%=
(L_1+L_2)(mathbf{P}_1 - mathbf{P}_2)^{-1}
leqslant
mathbb{Z}^r .
end{equation}
This concise description allows to use linear algebra to compute $S_{{1,2}}$, given
$L_1,L_2,mathbf{P}_1,mathbf{P}_2$.
Question:
I am interested in obtaining an analogous expression but with the intersection involving any finite (say $k$) number of cosets. More precisely, given $L_i$ subgroups of $mathbb{Z}^r$, and $mathbf{P}_i$ integer matrices of size $r times m$ $(iin I_k={1,ldots,k})$. I would like to obtain a "nice" description for the set
begin{equation}
S_{I_k}
=
left{
win mathbb{Z}^r :, bigcapnolimits_{i=1}^k (wmathbf{P}_i + L_i) neq varnothing
right},
end{equation}
in terms of the data $L_i,mathbf{P}_i$ that allows to compute it (or at least its index in $mathbb{Z}^r$).
Discussion of the case $k=3$.
Suppose that we want to describe the set:
begin{equation}
S_{{1,2,3}}
=
{
w in mathbb{Z}^r colon
(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) cap (wmathbf{P}_3 + L_3) neq varnothing
}.
end{equation}
Since the intersection of two cosets is either empty, or a coset of the intersection of the defining subgroups, one is tempted to
restrict the possible $w$'s in $S_{{1,2,3}}$ to $S_{{1,2}} = (L_1+L_2)(mathbf{P}_1 - mathbf{P}_2)^{-1}$ (since a triple intersection can only be nonempty if the double intersections are nonempty), write $(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2)$ as a coset of $L_1 cap L_2$ to reduce the triple intersection to a double intersection, and then reduce to the case $k=2$. Namely, write
begin{equation}
(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) cap (wmathbf{P}_3 + L_3)
=
(w' + (L_1 cap L_2) cap (wmathbf{P}_3 + L_3),
end{equation}
where w' is some element in the (now nonempty) intersection $(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2)$. The problem with this approach is that $w'$ depends on $w$ and so it does the exression we get for $S_{{1,2,3}}$, which, however, should depend only on $L_1,L_2,L_3,mathbf{P_1},mathbf{P_2},mathbf{P_3}$. Do you see a way (maybe another approach?) that allows to get rid of this dependence on w'?
Final remark.
Note that if the $L_i$'s are in direct sum (i.e., $sum_{iin I} L_i = bigoplus_{iin I} L_i$), then:
begin{equation}
bigcapnolimits_{i=1}^k (wmathbf{P}_i + L_i) neq varnothing
quad text{if and only if}quad
(wmathbf{P}_i + L_i) cap (wmathbf{P}_{i+1} + L_{i+1}) neq varnothing,
text{ for all } i text{ mod } k,
end{equation}
and hence, in this case $S_{I_k} = bigcap_{i text{ mod }k} S_{{i,i+1}}$, depending only in the initial data as we wanted.
Would it be possible to change the initial $L_i$'s by some $L'_i$'s in direct sum, such that the emptyness of the tested intersection does not change, and hence reduce the general case to the direct sum case?
linear-algebra matrices matrix-equations free-modules free-abelian-group
asked Jan 13 at 16:33
suitangisuitangi
45928
$endgroup$
Let $L_1,L_2$ be subgroups of $mathbb{Z}^m$, and let $mathbf{P}_1,mathbf{P}_2$ be $rtimes m$ integer matrices. Then, it is straightforward to check that the set
begin{equation}
S_{{1,2}} := {winmathbb{Z}^r colon (wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) neq varnothing}
end{equation}
is the set (indeed subgroup) of preimages by $mathbf{P}_1 - mathbf{P}_2$ of the subgroup $L_1 + L_2$. That is,
begin{equation} label{eq: 2-intersection}
S_{{1,2}}
=
%{winmathbb{Z}^r colon (wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) neq varnothing}
%=
(L_1+L_2)(mathbf{P}_1 - mathbf{P}_2)^{-1}
leqslant
mathbb{Z}^r .
end{equation}
This concise description allows to use linear algebra to compute $S_{{1,2}}$, given
$L_1,L_2,mathbf{P}_1,mathbf{P}_2$.
Question:
I am interested in obtaining an analogous expression but with the intersection involving any finite (say $k$) number of cosets. More precisely, given $L_i$ subgroups of $mathbb{Z}^r$, and $mathbf{P}_i$ integer matrices of size $r times m$ $(iin I_k={1,ldots,k})$. I would like to obtain a "nice" description for the set
begin{equation}
S_{I_k}
=
left{
win mathbb{Z}^r :, bigcapnolimits_{i=1}^k (wmathbf{P}_i + L_i) neq varnothing
right},
end{equation}
in terms of the data $L_i,mathbf{P}_i$ that allows to compute it (or at least its index in $mathbb{Z}^r$).
Discussion of the case $k=3$.
Suppose that we want to describe the set:
begin{equation}
S_{{1,2,3}}
=
{
w in mathbb{Z}^r colon
(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) cap (wmathbf{P}_3 + L_3) neq varnothing
}.
end{equation}
Since the intersection of two cosets is either empty, or a coset of the intersection of the defining subgroups, one is tempted to
restrict the possible $w$'s in $S_{{1,2,3}}$ to $S_{{1,2}} = (L_1+L_2)(mathbf{P}_1 - mathbf{P}_2)^{-1}$ (since a triple intersection can only be nonempty if the double intersections are nonempty), write $(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2)$ as a coset of $L_1 cap L_2$ to reduce the triple intersection to a double intersection, and then reduce to the case $k=2$. Namely, write
begin{equation}
(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2) cap (wmathbf{P}_3 + L_3)
=
(w' + (L_1 cap L_2) cap (wmathbf{P}_3 + L_3),
end{equation}
where w' is some element in the (now nonempty) intersection $(wmathbf{P}_1 + L_1) cap (wmathbf{P}_2 + L_2)$. The problem with this approach is that $w'$ depends on $w$ and so it does the exression we get for $S_{{1,2,3}}$, which, however, should depend only on $L_1,L_2,L_3,mathbf{P_1},mathbf{P_2},mathbf{P_3}$. Do you see a way (maybe another approach?) that allows to get rid of this dependence on w'?
Final remark.
Note that if the $L_i$'s are in direct sum (i.e., $sum_{iin I} L_i = bigoplus_{iin I} L_i$), then:
begin{equation}
bigcapnolimits_{i=1}^k (wmathbf{P}_i + L_i) neq varnothing
quad text{if and only if}quad
(wmathbf{P}_i + L_i) cap (wmathbf{P}_{i+1} + L_{i+1}) neq varnothing,
text{ for all } i text{ mod } k,
end{equation}
and hence, in this case $S_{I_k} = bigcap_{i text{ mod }k} S_{{i,i+1}}$, depending only in the initial data as we wanted.
Would it be possible to change the initial $L_i$'s by some $L'_i$'s in direct sum, such that the emptyness of the tested intersection does not change, and hence reduce the general case to the direct sum case?
linear-algebra matrices matrix-equations free-modules free-abelian-group
linear-algebra matrices matrix-equations free-modules free-abelian-group
asked Jan 13 at 16:33
suitangisuitangi
45928
asked Jan 13 at 16:33
suitangisuitangi
45928
edited Jan 16 at 18:32
suitangi
asked Jan 13 at 16:33
suitangisuitangi
45928
asked Jan 13 at 16:33
suitangisuitangi
45928
asked Jan 13 at 16:33
suitangisuitangi
45928
45928
add a comment |
add a comment |
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