##### This material has been published in
Rend. Sem. Mat. Univ. Padova **121**
(2009), 179-199,
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certified and accepted after peer review. Copyright and all rights therein
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without explicit permission.

##
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 *p*_{k}(*n*) the power
sum symmetric polynomial in *n* variables
*x*_{1}^{k}+*x*_{2}^{k}+...+*x*_{n}^{k}. 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 *p*_{a}(*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 *p*_{a}(3),*p*_{b}(3),p_{c}(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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