Number of terms in a symmetric polynomial

Algebra Level pending

If $P$ is a 4th degree symmetric polynomial in 3 variables, then what is the maximum number of terms in $P$ ?

The answer is 35.

1 solution

Andy Hayes
Apr 1, 2016

In a symmetric polynomial, each variable is interchangeable. If you exchange the values of two of the variables, then the resulting value of the polynomial will stay the same.

For example, if $Q(x,y)$ is a symmetric polynomial, then $Q(3,5)=Q(5,3)$ .

In order to obtain this effect, then all permutations of the variables in a term must be included, and the coefficients on these 'permuted' terms must be the same. For example, if there is an $2x^2y$ term in a symmetric polynomial, then there must also be a $2xy^2$ term.

For the given problem, the goal is to obtain all possible 4th-and-below degree terms, and find the number of permutations of the variables in each term. Below is a chart showing this analysis.

Degree of term	# of Variables in Term	Format of Term	Number of Permutations
4	3	$x^2yz$	3
4	2	$x^3y$	6
4	2	$x^2y^2$	3
4	1	$x^4$	3
3	3	$xyz$	1
3	2	$x^2y$	6
3	1	$x^3$	3
2	2	$xy$	3
2	1	$x^2$	3
1	1	$x$	3
0	0	$1$	1

Adding up all the possible numbers of terms yields $\boxed{35}$ .

Number of terms in a symmetric polynomial

The answer is 35.

1 solution

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