Mathematics 567
Homework (due Apr 4)
A. Hulpke
41) Let F be a field and x1 , . . . , x n be indeterminates. A polynomial f ∈ F[x1 , . . . , x n ] is called
symmetric, if for any permutation σ of the indeterminates we have that σ( f ) = f .
Obviously,the following functions (called elementary symmetric polynomials) are symmetric:
S1 (x1 , . . . , x n ) = x1 + x2 + x3 + ⋯ + x n
S2 (x1 , . . . , x n ) = x1 x2 + x1 x3 + ⋯ + x n−1 x n
S3 (x1 , . . . , x n ) = ∑ x1 x j x k
i< j<k
⋮
S n (x1 , . . . , x n ) = x1 x2 ⋅ ⋯ ⋅ x n
The aim of this exercise is to prove the fundamental theorem of elementary symmetric polynomials:
Theorem: Let f ∈ F[x1 , . . . , x n ] symmetric, then exists g ∈ F[x1 , . . . , x n ] such that
f (x1 , . . . , x n ) = g (S1 (x1 , . . . , x n ), S2 (x1 , . . . , x n ), . . . , S n (x1 , . . . , x n )) .
Example: x12 x22 x3 + x12 x2 x32 + x1 x22 x32 + x12 + 2x1 x2 + 2x1 x3 + x22 + 2x2 x3 + x32 = S12 + S2 ∗ S3 .
We define an ordering on monomials by setting: x1d1 x2d2 ⋯x nd n > y1e1 y2e2 ⋯y ne n if and only if for some i:
d j = e j for j < i and d i > e i (lexicographic comparison of the exponent vectors).
a) Suppose that d1 ≥ d2 ≥ ⋯ ≥ d n is a sequence of integers. Show that x1d1 x2d2 ⋯x nd n is the largest (with
respect to this ordering) term of the polynomial
d n−2 −d n−1 −d n
d n−1 −d n
d n−1 −d n
Td1 ,...,d n (x1 , . . . , x n ) ∶= S1d1 −d2 −d3 −⋯−d n ⋅ ⋯ ⋅ S n−2
⋅ S n−1
⋅ S n−1
⋅ S nd n
b) Suppose that f (x1 , . . . , x n ) is symmetric and c ⋅ x1d1 x2d2 ⋯x nd n is the largest (with respect to this
ordering) term of f and let Td1 ,...,d n (x1 , . . . , x n ) for these d i as in a). Show that the largest monomial
of f (x1 , . . . , x n ) − c ⋅ Td1 ,...,d n (x1 , . . . , x n ) is strictly smaller than x1d1 x2d2 ⋯x nd n .
c) Show that by iterating the process in b) you can build the polynomial g as claimed in the theorem.
d) Express
f
= x 2 y + x 2 z + 3x yz + y 2 x + y 2 z + z 2 x + yz 2 − 2x 3 y 3 z
−4x 3 y 2 z 2 − 4x 2 y 3 z 2 − 2x 3 z 3 y − 4x 2 y 2 z 3 − 2x y 3 z 3
as a polynomial in the elementary symmetric polynomials in x, y and z.