# nt.quantity concept – Diophantine equation in regards to the automorphism group of lattice by constraints Answer

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## nt.quantity concept – Diophantine equation in regards to the automorphism group of lattice by constraints

Fixed $$sigma_x = left ( commence {array} {cc} 0 & 1 1 & 0 aim {array} privilege)$$ and $$Ok = left ( commence {array} {ccc} 3 & 32 & -64 1 & 32 & -32 -2 & -32 & 64 aim {array} privilege),$$

I wish to discover $$n$$ in order that, $$K_n = sigma_x oplus sigma_x oplus .. sigma_x oplus Ok$$. and the corresponding $$W in GL (2n + 3, mathbb {Z})$$ in order that $$W ^ {T} K_n ^ {- 1} W = K_n ^ {- 1},$$ the one restrictions on $$W$$ the privilege nook ought to breathe $$left ( commence {array} {cc} 8 & -5 5 & 13 aim {array} privilege).$$ For instance when $$n = 1,$$ I cerebrate $$W$$ appears to be like love $$W = commence {bmatrix} a_1 & a_2 & a_3 & a_4 & a_5 b_1 & b_2 & b_3 & b_4 & b_5 c_1 & c_2 & c_3 & c_4 & c_5 d_1 & d_2 & d_3 & 8 & – 5 e_1 & e_2 & e_3 & 5 & 13 aim {bmatrix}$$ in order that:
$$commence {equation} W ^ {T} K_1 ^ {- 1} W = K_1 ^ {- 1}. aim {equation}$$
I’m making an attempt to employ Mathematica and may’t discover any options. Then I strive $$n = 2$$ and I quiet cannot discover a resolution. But I do not know methods to show that there isn’t a resolution for $$W ^ {T} K_n ^ {- 1} W = K_n ^ {- 1}.$$ for every $$n$$. I cerebrate there ought to breathe $$1$$.

My observation on this downside:

1 I can not discover mighty literature on fixing the Diophantine equation in integer quadratic figure (at the least for $$4 occasions 4$$). But my fantasy is that when $$n$$ grows, there ought to breathe extra liberty and extra variables. So you need to breathe capable of finding an answer, or at the least show that the answer exists.

2 pages $$K_n oplus sigma_x$$ is an indeterminate matrix, so there isn’t a computational system to seek out this automorphism group of $$K_n$$. See associated query right here. An indeterminate matrix results in an innumerable automorphism group, so the capability of the automorphism group ought to breathe sizable sufficient to accommodate such $$W$$.

3 Notice that Smith’s regular figure for $$= left ( commence {array} {ccc} 1 & 0 & 0 0 & 32 & 0 0 & 0 & 32 aim {array} privilege).$$ So for everybody $$K_n$$ it all the time has two nontrivial elements $$32$$. So there’s a limitation for $$2 occasions 2$$ Corner for $$W$$.

My analysis has an issue associated to this kind of given query $$Ok$$ and $$W$$ (limit $$W$$ for the scowl privilege nook and pretense that yow will discover sufficiently big $$Ok$$ (extend $$Ok$$ and $$W$$) with the identical and nontrivial Smith regular ($$neq 1$$) train and answer $$W ^ {T} K_1 ^ {- 1} W = K_1 ^ {- 1}.$$ )

Comments and outcomes are very welcome, thanks very mighty.

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