Worked solution: Hilbert's existence proof for Waring's problem (1909)

Step 3 of 7: Hilbert's identity: a sum of squares to the

k

is a combination of

2k

-th powers

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In plain words

Here is the single algebraic engine driving the whole proof. Hilbert shows that for every $k$ and every number of variables $r$ , there exist finitely many positive rational numbers $a_i$ and integer coefficients $b_{i,1},\dots,b_{i,r}$ so that the polynomial $(x_1^2+\cdots+x_r^2)^k$ — degree $2k$ , written using squares — equals a fixed combination of $2k$ -th powers of linear combinations of the $x$ 's.

This is remarkable because the left side looks nothing like a sum of powers of linear forms, yet it can always be rewritten as exactly that, for a suitable (large but fixed) number $M$ of terms. Classical special cases of this had already been found by hand: for $r=4$ variables, $k=2$ gives Liouville's identity for fourth powers, $k=3$ gives an identity due to Fleck for sixth powers, and $k=4$ gives one due to Hurwitz for eighth powers — Hilbert's contribution was to show such an identity exists for every $k$ at once, and to explain, in general, where it comes from.

(x_1^2+\cdots+x_r^2)^k = \sum_{i=1}^{M} a_i\,(b_{i,1}x_1+\cdots+b_{i,r}x_r)^{2k}

Detailed analysis

Hilbert's identity (as stated in course presentations of Nathanson's proof, e.g. Baxter's 'Additive Number Theory Seminar' notes, Theorem 1) reads: for every $k\ge1$ and $r\ge1$ there exist an integer $M$ , positive rational numbers $a_1,\dots,a_M$ , and integers $b_{i,j}$ ( $i=1,\dots,M$ , $j=1,\dots,r$ ) such that $(x_1^2+\cdots+x_r^2)^k=\sum_{i=1}^M a_i(b_{i,1}x_1+\cdots+b_{i,r}x_r)^{2k}$ identically in the real variables $x_1,\dots,x_r$ .

Classical hand-built instances of exactly this shape predate Hilbert: with $r=4$ , Liouville's $k=2$ identity $6(x_1^2+x_2^2+x_3^2+x_4^2)^2=\sum_{i<j}(x_i+x_j)^4+\sum_{i<j}(x_i-x_j)^4$ gives $g(4)\le53$ ; Fleck's $k=3$ identity involves sixth powers of $x_i\pm x_j\pm x_l$ , $x_i\pm x_j$ , and $x_i^6$ ; Hurwitz's $k=4$ identity involves eighth powers. Each is a special case of Hilbert's general identity for the specific pair $(k,r=4)$ .

The existence of such an identity for arbitrary $k$ can be understood via averaging: consider the average value of $(u\cdot x)^{2k}=(u_1x_1+\cdots+u_rx_r)^{2k}$ as $u=(u_1,\dots,u_r)$ ranges uniformly over the unit sphere $\|u\|=1$ in $\mathbb{R}^r$ . By rotational symmetry this average depends only on $\|x\|^{2k}=(x_1^2+\cdots+x_r^2)^k$ , giving a continuous integral identity $(x_1^2+\cdots+x_r^2)^k=c_{r,k}\int_{\|u\|=1}(u\cdot x)^{2k}\,d\sigma(u)$ for an explicit positive constant $c_{r,k}$ . Hilbert's algebraic identity is a finite, discretized version of this continuous average — replacing the sphere integral with a finite weighted sum over finitely many directions $u^{(i)}=(b_{i,1},\dots,b_{i,r})$ with rational weights $a_i$ , in a way that reproduces the integral exactly for this particular degree- $2k$ integrand (a classical device in the theory of spherical designs / cubature formulas).

Terms in this step

Linear form: An expression $b_1x_1+\cdots+b_rx_r$ that is a fixed linear combination of the variables $x_1,\dots,x_r$ with constant coefficients $b_1,\dots,b_r$ — the simplest possible 'shape' of a multivariable expression, of degree $1$ .
Averaging over the unit sphere: Integrating a function of a direction vector $u$ against the uniform (rotation-invariant) probability measure on the unit sphere $\|u\|=1$ ; because this measure is invariant under rotations, the resulting average can only depend on rotation-invariant features of the other data, such as $\|x\|$ .