Finding solutions with distinct variables to systems of linear equations over $$\mathbb {F}_p$$ F p
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208_2022_Article_2391.pdf
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523.9 KB
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Author(s)
Sauermann, Lisa
Date Issued
April 1, 2022
Publisher
Springer Berlin Heidelberg
Citation
Sauermann, Lisa. 2022. "Finding solutions with distinct variables to systems of linear equations over $$\mathbb {F}_p$$ F p."
Version
Final published version
Abstract
Abstract
Let us fix a prime p and a homogeneous system of m linear equations
$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$
a
j
,
1
x
1
+
⋯
+
a
j
,
k
x
k
=
0
for
$$j=1,\dots ,m$$
j
=
1
,
⋯
,
m
with coefficients
$$a_{j,i}\in \mathbb {F}_p$$
a
j
,
i
∈
F
p
. Suppose that
$$k\ge 3m$$
k
≥
3
m
, that
$$a_{j,1}+\dots +a_{j,k}=0$$
a
j
,
1
+
⋯
+
a
j
,
k
=
0
for
$$j=1,\dots ,m$$
j
=
1
,
⋯
,
m
and that every
$$m\times m$$
m
×
m
minor of the
$$m\times k$$
m
×
k
matrix
$$(a_{j,i})_{j,i}$$
(
a
j
,
i
)
j
,
i
is non-singular. Then we prove that for any (large) n, any subset
$$A\subseteq \mathbb {F}_p^n$$
A
⊆
F
p
n
of size
$$|A|> C\cdot \Gamma ^n$$
|
A
|
>
C
·
Γ
n
contains a solution
$$(x_1,\dots ,x_k)\in A^k$$
(
x
1
,
⋯
,
x
k
)
∈
A
k
to the given system of equations such that the vectors
$$x_1,\dots ,x_k\in A$$
x
1
,
⋯
,
x
k
∈
A
are all distinct. Here, C and
$$\Gamma $$
Γ
are constants only depending on p, m and k such that
$$\Gamma
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00208-022-02391-y
