Solving OR bounding sums of solutions to certain linear diophantine equations

Solving OR bounding sums of solutions to certain linear diophantine equations

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(This question arose during group work on classifying modular tensor categories.)

Let $p$ and $q$ be two distinct primes. We seek integral solutions $lbrace x_i,j rbrace$ to the equation

beginalign
0 = sum_i=0^i=m sum_j=m-i^j=m p^i q^j x_i,j
endalign
subject to the requirements

beginalign
x_m,m=-1, \ p nmid x_i,j, \ q nmid x_i,j.
endalign

For example, when $p=5$, $q=7$, and $m=3$, we want to find integral solutions to the equation

beginalign
5^3 7^3 = 5^1 7^2 x_1,2 + 5^2 7^1 x_2,1 + 5^3 7^0 x_3,0 + 5^0 7^3 x_0,3 + 5^2 7^2 x_2,2 + 5^3 7^1 x_3,1 + 5^1 7^3 x_1,3 + 5^3 7^2 x_3,2 + 5^2 7^3 x_2,3
endalign

such that none of the non-zero $x_i,j$ are divisible by 5 or 7. (We do allow $x_i,j$ to equal $0$.)

Question 1. What algebraic geometry would be necessary to attack this problem? It seems related to "finding integral points," so it could be much more difficult than I appreciate.

Question 2. If determining integral solutions is too hard: Is there a way we could at least give a bound for the sum of all the $x_i,j$ in terms of $m$? That is, can we say

beginalign
sum_i=0^i=m sum_j=m-i^j=m x_i,j , < , f(m)
endalign

for some function $f(m)$ that doesn't grow too fast?

edited Jul 20 at 20:19

asked Jul 20 at 16:36

L.C. Ruth

513

add a commentÂ |Â

up vote
1
down vote

favorite

(This question arose during group work on classifying modular tensor categories.)

Let $p$ and $q$ be two distinct primes. We seek integral solutions $lbrace x_i,j rbrace$ to the equation

beginalign
0 = sum_i=0^i=m sum_j=m-i^j=m p^i q^j x_i,j
endalign
subject to the requirements

beginalign
x_m,m=-1, \ p nmid x_i,j, \ q nmid x_i,j.
endalign

For example, when $p=5$, $q=7$, and $m=3$, we want to find integral solutions to the equation

beginalign
5^3 7^3 = 5^1 7^2 x_1,2 + 5^2 7^1 x_2,1 + 5^3 7^0 x_3,0 + 5^0 7^3 x_0,3 + 5^2 7^2 x_2,2 + 5^3 7^1 x_3,1 + 5^1 7^3 x_1,3 + 5^3 7^2 x_3,2 + 5^2 7^3 x_2,3
endalign

such that none of the non-zero $x_i,j$ are divisible by 5 or 7. (We do allow $x_i,j$ to equal $0$.)

Question 1. What algebraic geometry would be necessary to attack this problem? It seems related to "finding integral points," so it could be much more difficult than I appreciate.

Question 2. If determining integral solutions is too hard: Is there a way we could at least give a bound for the sum of all the $x_i,j$ in terms of $m$? That is, can we say

beginalign
sum_i=0^i=m sum_j=m-i^j=m x_i,j , < , f(m)
endalign

for some function $f(m)$ that doesn't grow too fast?

edited Jul 20 at 20:19

asked Jul 20 at 16:36

L.C. Ruth

513

add a commentÂ |Â

up vote
1
down vote

favorite

up vote
1
down vote

favorite

(This question arose during group work on classifying modular tensor categories.)

Let $p$ and $q$ be two distinct primes. We seek integral solutions $lbrace x_i,j rbrace$ to the equation

beginalign
0 = sum_i=0^i=m sum_j=m-i^j=m p^i q^j x_i,j
endalign
subject to the requirements

beginalign
x_m,m=-1, \ p nmid x_i,j, \ q nmid x_i,j.
endalign

For example, when $p=5$, $q=7$, and $m=3$, we want to find integral solutions to the equation

beginalign
5^3 7^3 = 5^1 7^2 x_1,2 + 5^2 7^1 x_2,1 + 5^3 7^0 x_3,0 + 5^0 7^3 x_0,3 + 5^2 7^2 x_2,2 + 5^3 7^1 x_3,1 + 5^1 7^3 x_1,3 + 5^3 7^2 x_3,2 + 5^2 7^3 x_2,3
endalign

such that none of the non-zero $x_i,j$ are divisible by 5 or 7. (We do allow $x_i,j$ to equal $0$.)

Question 1. What algebraic geometry would be necessary to attack this problem? It seems related to "finding integral points," so it could be much more difficult than I appreciate.

Question 2. If determining integral solutions is too hard: Is there a way we could at least give a bound for the sum of all the $x_i,j$ in terms of $m$? That is, can we say

beginalign
sum_i=0^i=m sum_j=m-i^j=m x_i,j , < , f(m)
endalign

for some function $f(m)$ that doesn't grow too fast?

edited Jul 20 at 20:19

asked Jul 20 at 16:36

L.C. Ruth

513

(This question arose during group work on classifying modular tensor categories.)

Let $p$ and $q$ be two distinct primes. We seek integral solutions $lbrace x_i,j rbrace$ to the equation

beginalign
0 = sum_i=0^i=m sum_j=m-i^j=m p^i q^j x_i,j
endalign
subject to the requirements

beginalign
x_m,m=-1, \ p nmid x_i,j, \ q nmid x_i,j.
endalign

For example, when $p=5$, $q=7$, and $m=3$, we want to find integral solutions to the equation

beginalign
5^3 7^3 = 5^1 7^2 x_1,2 + 5^2 7^1 x_2,1 + 5^3 7^0 x_3,0 + 5^0 7^3 x_0,3 + 5^2 7^2 x_2,2 + 5^3 7^1 x_3,1 + 5^1 7^3 x_1,3 + 5^3 7^2 x_3,2 + 5^2 7^3 x_2,3
endalign

such that none of the non-zero $x_i,j$ are divisible by 5 or 7. (We do allow $x_i,j$ to equal $0$.)

Question 1. What algebraic geometry would be necessary to attack this problem? It seems related to "finding integral points," so it could be much more difficult than I appreciate.

Question 2. If determining integral solutions is too hard: Is there a way we could at least give a bound for the sum of all the $x_i,j$ in terms of $m$? That is, can we say

beginalign
sum_i=0^i=m sum_j=m-i^j=m x_i,j , < , f(m)
endalign

for some function $f(m)$ that doesn't grow too fast?

edited Jul 20 at 20:19

asked Jul 20 at 16:36

L.C. Ruth

513

edited Jul 20 at 20:19

edited Jul 20 at 20:19

edited Jul 20 at 20:19

asked Jul 20 at 16:36

L.C. Ruth

513

asked Jul 20 at 16:36

L.C. Ruth

513

asked Jul 20 at 16:36

L.C. Ruth

513

513

add a commentÂ |Â

add a commentÂ |Â

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