# Tagged Questions

A lattice in $\mathbb R^n$ is a discrete subgroup of $\mathbb R^n$ or, equivalently, it is a subgroup of $\mathbb R^n$ generated by linearly independent vectors. Lattices have applications in geometric number theory, e.g. via Minkowski's theorem.

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### Ehrhart Polynomials Modulo Prime Integers

Are there any results known about computing Ehrhart Polynomials modulo prime integers?
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### $(a \vee b)\wedge c=b\wedge c$ implies $(c\wedge b)\wedge a= b \vee c$

Show that for any elements a,b,c in a modular lattice $(a \vee b)\wedge c=b\wedge c$ implies $(c\wedge b)\wedge a= b \vee c$ ? $\wedge$ is meet and $\vee$ is join operations respectively .
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### “All vertices of a convex pentagon are lattice points, and its sides have integral length. Show that its perimeter is even.”

"All vertices of a convex pentagon are lattice points, and its sides have integral length. Show that its perimeter is even." - Problem Solving Strategies, Arthur Engel, pg. 27. I have proven the ...
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### Minimum difference of angles between points on square lattice

I have integer grid of size $N \times N$. If I calculate angles between all point triples - is it possible analytically find minimal non-zero difference between those angles?
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### Are there 2D analogues for integer division and modular arithmetic?

Let's say you have a "parallelogram" of points $P = \{(0, 0), (0, 1), (1, 1), (0, 2), (1, 2)\}$. This parallelogram lies between $u = (2, 1)$ and $v = (-1, 2)$. Then for any point $n \in \mathbb{Z}^2$...
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### show that L^+ is non-empty

I want to show that $L^+$ is non-empty where $L$ is a full-rank integer lattice and $L^+$ denotes the set of elements of $L$ having positive coordinates I have an indication that I did not understand ...
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### Algebraic proof of Ehrhart's theorem

Let $P \subset \mathbb{R}^d$ be a $d$-dimensional polytope, where all vertices lie on integral coordinates, and let $L(P,n)$ denote the number of integral lattice points contained in the scaled ...
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### moduli of lattices

Consider the set $M$ of all (rank $g$) lattices in $g$-dimensional complex affine space $C^g$. Does M identify in some way with Siegel upper half space $H_g$? Let's say a lattice has CM if it has ...
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### Lattices and Elliptic curves and number fields

Let $K$ be a number field with ring of integers $O_K$. If $K$ is totally real, then $O_K$ is a lattice in $\mathbb R$. If $K$ is imaginary quadratic, then $O_K$ is a lattice in $\mathbb C$. If $K$ ...
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### smallest integer contained in sublattice $\Rightarrow$ $L'=[q,r\tau+s]$

Let $L'$ be a sublattice of the lattice $[1,\tau]$ in a imaginary quadratic field. Reminder: a lattice $L$ consists of the $\mathbb{Z}$-linear combinations of $1$ and $\tau$, with $\{1,\tau \}$ linear ...
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### Helping solve mod problems

I am having trouble solving the below problems. My teacher taught us to write out the solutions by hand.. but I really think there is an easier way to do the higher numbers. Thanks!
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### Counting sum of lattice points

Assume a set $S$ with $|S|$ entries. Indeed, $S$ is the set of lattice points inside a $k$-sphere. Assume $V=S\oplus S$ where $\oplus$ is the Minkowski sum of two sets. Do you know any lower bound on ...
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### Is this the subgroup lattice for $\Bbb{Z}_4 \times \Bbb{Z}_8$?

I have been attempting to create the subgroup lattice for $\Bbb{Z}_4 \times \Bbb{Z}_8$. I have, so far, this: http://www.scribd.com/doc/223680804/Subgroup-Lattice-of-Z-4-x-Z-8 While I have calculated ...
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### Correctness of the size of an planar integer lattice unknot

A planar integer lattice unknot is a polygon drawn over a two dimensional integer lattice. Here is an example: Given a number $N$, a planar unknot is not always possible. For example, a planar ...
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### Integer polynomial with two aligned roots

Anyone have an example of a monic polynomial $f(x) \in \mathbb{Z}[x]$ with $f(0)=1$ such that for $\alpha \in \mathbb{C}\backslash\mathbb{R}$ and $t \in R, t>0, t\neq 1$ both $\alpha$ and $t\alpha$ ...
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### number of lattice points in an n-ball

I have faced a problem in my work and I will appreciate any hint/reference as I am not much into the lattice problems. Assume an n-dimensional lattice $\Lambda_n$ with generator matrix $G$. Note that ...
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### Lattice Definition

I see two Lattice definitions in Mathematics. Partial order set with each pair of elements have a least least upper bound and greatest lower bound. Integer linear combinations of vectors. Is ...
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### How to count lattice points on a line.

How can we count the number of lattice point on a line, given that the endpoints of the lines are themselves lattice points? I really can't think of how counting lattice points would work, so please ...
### Show $\binom{n}{k}\binom{k}{a} = \binom{n}{a}\binom{n-a}{k-a}$ by block-walking interpretation of Pascal's triangle
A combinatorial proof for the identity $$\binom{n}{k}\binom{k}{a} = \binom{n}{a}\binom{n-a}{k-a}$$ is the following "committee" argument, which seems the most common. There are $\binom{n}{k}$ ...