Are two sequences equal if the sums and sums of squares are equal?

Are two sequences $(x_i)_{i=1,\ldots,n}$ and $(y_i)_{i=1,\ldots,n}$ equal if $\sum_{i=1}^nx_i=\sum_{i=1}^ny_i$ and $\sum_{i=1}^nx_i^2=\sum_{i=1}^ny_i^2$?

Solution 1:

No.

$$ x = (1,1,1,-3) $$ $$ y = (-1,-1,-1,3) $$

$$\sum x_i = \sum y_i = 0$$ $$ \sum x_i^2 = \sum y_i^2 =12$$

Solution 2:

No! You need $n$ polynomial equations.

If I remember correctly, given a set of $n$ distinct real numbers $\mathcal Y :=\{y_1, y_2, \dots, y_n\}$, the following system of $n$ polynomial equations

$$\begin{array}{rl} x_1 + x_2 + \dots + x_n &= c_1\\ x_1^2 + x_2^2 + \dots + x_n^2 &= c_2\\ \vdots \\ x_1^n + x_2^n + \dots + x_n^n &= c_n\end{array}$$

where

$$c_k := y_1^k + y_2^k + \dots + y_n^k$$

has $n!$ solutions, namely, all $n!$ permutations of the $n$ distinct elements of $\mathcal Y$. One can use algebraic geometry to prove this. I vaguely recall this being related to moment problems.

Example

Let $\mathcal Y := \{1,3,6\}$. Intersecting the plane with the sphere,

$$\begin{array}{rl} x_1 + x_2 + x_3 &= 10\\ x_1^2 + x_2^2 + x_3^2 &= 46\end{array}$$

we obtain a circle on the plane, as depicted below

However, intersecting these two with the cubic surface,

$$\begin{array}{rl} x_1 + x_2 + x_3 &= 10\\ x_1^2 + x_2^2 + x_3^2 &= 46\\ x_1^3 + x_2^3 + x_3^3 &= 244\end{array}$$

we obtain the $3!=6$ permutations of the elements of $\mathcal Y$, which are colored in red

If we plot the quadratic and the cubic surfaces, but not the plane,

and, from another point of view, we have what looks like a stylish hat

Solution 3:

To consider an even more basic counterexample. What happens if you simply reorder your sequence? Take $x = (0,1)$ and $y = (1,0)$. (I assume you're talking about sequences, so above $x \neq y$; of course, as sets, $\{0,1\} = \{1,0\}$.)