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Let $A$ be a symmetric $d\times d$ matrix with integer entries such that the quadratic form $Q(x)=\langle Ax,x\rangle, x\in \mathbb{R}^d$, is non-negative definite. For which $d$ does it imply that $Q$ is a sum of finitely many squares of linear forms with integer coefficients $$ Q(x)=\sum_{i=1}^N (\ell_i(x))^2\quad \text{for some}\, N? $$ For $d=2$ this is true, I know it from the problem proposed by Sweden to IMO in 1995, but probably all this stuff is known for a longer time.

I think, I may prove it for some other small dimensions, although not so elementary (using Minkowski theorem on lattice points in convex bodies: if $Q$ is positive definite, we may find a linear form $\ell(x)$ such that $Q-\ell^2$ is still non-negative definite, this is equivalent to finding an integer point in an ellipsoid), but for large $d$ this argument fails.

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  • $\begingroup$ This is related to Hilbert 17th problem, right? en.wikipedia.org/wiki/Hilbert%27s_seventeenth_problem $\endgroup$ – Francesco Polizzi Apr 12 '19 at 10:25
  • $\begingroup$ @FrancescoPolizzi well, in a sense both questions generalize the classical statement "non-negative definite real quadratic form is a sum of squares of real linear functions": Hilbert 17th problem extends it to non-negative polynomials of higher degree, this question for integer (not just real) coefficients. $\endgroup$ – Fedor Petrov Apr 12 '19 at 10:48
  • $\begingroup$ I think this is answered at mathoverflow.net/questions/9073/… $\endgroup$ – Gerry Myerson Apr 12 '19 at 12:18
  • $\begingroup$ @GerryMyerson I do not see the answer, would you please speciy where? $\endgroup$ – Fedor Petrov Apr 12 '19 at 14:39
  • $\begingroup$ Sorry, Fedor, I got confused between higher degree and more variables. $\endgroup$ – Gerry Myerson Apr 13 '19 at 4:18
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This is a well-known problem, called the Waring's problem of integral quadratic forms. Every semi-positive definite quadratic form in $n \leq 5$ variables is a sum of $n + 3$ squares of linear forms. This was proved by Chao Ko, but this can be explained by the fact that the quadratic form of sum of $n + 3$ squares has class number 1 if $n \leq 5$.

There are positive definite quadratic forms in $n \geq 6$ variables which cannot be written as sums of squares of integral linear forms. The smallest example is the quadratic form corresponding to the root system $E_6$.

So, one should look at the set of positive definite quadratic forms in $n$ variables that can be written as sums of squares of linear forms. Then there exists an integer $g(n)$ such that all these quadratic forms can be written as a sum of $g(n)$ squares of integral linear forms. The magnitude of $g(n)$ is not known. The best upper bound is $O(e^{k\sqrt{n}})$ for some explicit $k$. This is obtained recently by Beli-Chan-Icaza-Liu (appeared in TAMS).

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  • $\begingroup$ Thank you very much! I am bit confused about "the quadratic form of sum of $n+3$ squares has class number 1 if $n\le 5$". Which quadratic form in how many variables do we consider? $\endgroup$ – Fedor Petrov Apr 13 '19 at 4:45
  • $\begingroup$ I am talking about the quadratic form $x_1^2 + \cdots + x_{n+3}^2$. Its class number is 1 when $n \leq 5$. $\endgroup$ – WKC Apr 13 '19 at 5:58
  • $\begingroup$ ok, and how does it explain that a non-negative definite integer quadratic form in $n$ variables is a sum of $n+3$ squares? $\endgroup$ – Fedor Petrov Apr 13 '19 at 6:10
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    $\begingroup$ My apology! Let $A$ and $B$ be the Gram matrices of two integral quadratic forms. Suppose that $A$ is $\ell \times \ell$ and $B$ is $n \times n$ with $\ell \geq n$. We say that $B$ is represented by $A$ over a ring extension $R$ of $\mathbb Z$ if there exists an $\ell \times n$ matrix $T$ with entries from $R$ such that $T^t A T = B$. We say that $B$ is represented by the genus of $A$ if $B$ is represented by $A$ over $\mathbb R$ and over the ring of $p$-adic integers $\mathbb Z_p$ for every prime $p$. $\endgroup$ – WKC Apr 13 '19 at 21:35
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    $\begingroup$ We would like to have a local-global principle, i.e. if $B$ is represented by the genus of $A$ then $B$ is represented by $A$. But such principle does not hold in general, but if the class number of $A$ is 1, then we do have this local-global principle. $\endgroup$ – WKC Apr 13 '19 at 21:38
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Such problems were investigated by L. J. Mordell in the 1930s. Two related papers of Mordell are as follows:

  1. L. J. Mordell, A new Warings problem with squares of linear forms, Quarterly J. (Oxford series) 1 (1930), 276–288.

  2. L. J. Mordell, On binary quadratic forms expressable as a sum of three linear squares with integer coefficients, J. Reine Angew. Math. 167 (1932), 12–19.

For an introduction to Mordell's results on sums of squares of linear forms, see Section 2 of D. W. Hoffmann's recent preprint Sums of integers and sums of their squares where the author applied Mordell's results to study sums of squares with certain linear restrictions.

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I should add that there have been many results on this problem since the 1990s. The existence of the number $g(n)$ was established by Maria Icaza in her Ohio State 1992 thesis. You should check out her papers. A bit later Myung Kwan Kim and Byeong Kweon Oh published a few papers. They showed that $g(6) = 10$ (not 9 as Mordell and Ko thought in the 1930s) and this is the last known exact value of the function $g(n)$.

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  • $\begingroup$ This remark is better added to your other answer. $\endgroup$ – Greg Martin Apr 13 '19 at 7:07

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