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Miklós Schweitzer
2011 Miklós Schweitzer
5
5
Part of
2011 Miklós Schweitzer
Problems
(1)
linear algebra
Source: miklos schweitzer 2011 q5
8/29/2021
Let n, k be positive integers. Let
f
a
(
x
)
:
=
∣
∣
x
−
a
∣
∣
2
n
f_a(x) := ||x - a||^{2n}
f
a
(
x
)
:=
∣∣
x
−
a
∣
∣
2
n
, where the vectors
x
=
(
x
1
,
.
.
.
,
x
k
)
,
a
∈
R
k
x = (x_1, ..., x_k) , a\in R^k
x
=
(
x
1
,
...
,
x
k
)
,
a
∈
R
k
, and ||·|| is the Euclidean norm. Let the vector space
Q
n
,
k
Q_{n, k}
Q
n
,
k
be generated by the functions
f
a
f_a
f
a
(
a
∈
R
k
a\in R^k
a
∈
R
k
). What is the largest integer N for which
Q
n
,
k
Q_{n, k}
Q
n
,
k
contains all polynomials of
x
1
,
.
.
.
,
x
k
x_1, ..., x_k
x
1
,
...
,
x
k
whose total degree is at most N?
linear algebra
polynomial