This material has been published in
Rend. Sem. Mat. Univ. Padova 121
(2009), 179-199,
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Aldo Conca,
Christian Krattenthaler and
Junzo Watanabe
Regular sequences of symmetric polynomials
(17 pages)
Abstract.
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
only if
abc=0 (mod 6).
We provide evidence for the conjecture by proving it in several
special instances.
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