This material has been published in
Rend. Sem. Mat. Univ. Padova 121
the only definitive repository of the content that has been
certified and accepted after peer review. Copyright and all rights therein
are retained by European Mathematical Society Publishing House.
This material may not be copied or reposted
without explicit permission.
Christian Krattenthaler and
Regular sequences of symmetric polynomials
A set of n homogeneous polynomials in n variables is called a
regular sequence if the associated polynomial system has only the
obvious solution (0,0,...,0). Denote by pk(n) the power
sum symmetric polynomial in n variables
x1k+x2k+...+xnk. The interpretation of the q-analogue
of the binomial coefficient as Hilbert function leads us to
discover that n consecutive power sums in n variables form a
regular sequence. We consider then the following problem: describe
the subsets A of N* of cardinality n such that the set
of polynomials pa(n) with a \in A is a regular sequence.
We prove that a necessary condition is that n! divides the product
of the degrees of the elements of A. To find a sufficient condition
turns out to be surprisingly difficult already for n=3. Given
positive integers a<b<c with gcd(a,b,c)=1,
we conjecture that pa(3),pb(3),pc(3) is a regular sequence if and
abc=0 (mod 6).
We provide evidence for the conjecture by proving it in several
The following versions are available:
Back to Christian Krattenthaler's