A quadratic form is a polynomial $p(x_1,\dots,x_n)$ of the form $$ p(x_1,\dots,x_n)=\sum_{i \leq j}a_{ij}x_ix_j. $$ For example, $p_1(x,y,z,w)=x^2+y^2+z^2+w^2$ and $3x^2-5y^2$ are quadratic forms. I'm not sure this is relevant here, but for any such $p$ there is a symmetric matrix $A$ such that $p(\vec x)= (\vec x)^T A(\vec x)$, where we think of $\vec x$ as a column vector.

$p$ is positive definite iff $p(\vec x)>0$ whenever $\vec x\ne 0$.

The form is integer valued iff $p(\vec a)\in{\mathbb Z}$ whenever $\vec a\in {\mathbb Z}^n$.

For short, let's say "form" instead of "positive definite integer valued quadratic form". A form is universal iff it represents every positive integer, i.e., for all $m>0$ there is $\vec a\in{\mathbb Z}^n$ with $p(\vec a)=m$. For example, $p_1$ above is universal.

I read that no form in 3 or fewer variables is universal, and I was hoping for a proof (or at least a reference if the proof is too long/convoluted for it to be written here). I was also wondering whether the requirement of positive-definiteness is relevant here, and whether this is only for forms $p$ where the matrix $A$ above has integer entries.

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    $\begingroup$ The positive-definiteness is certainly relevant: for example, $x^2 + y^2 - z^2$ is an indefinite universal form in three variables. $\endgroup$ – Qiaochu Yuan May 17 '11 at 15:24
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    $\begingroup$ Positive definiteness is definitely relevant: the form $xy$ represents everything. $\endgroup$ – André Nicolas May 17 '11 at 15:27
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    $\begingroup$ May I suggest that you change the title of the question into something a bit more descriptive? "Quadratic forms" is rather vague. $\endgroup$ – Hans Lundmark May 17 '11 at 16:05
  • $\begingroup$ @Qiaochu, user6312: Thanks, I should have noticed this. @Hans: You are right, I edited the title. $\endgroup$ – Bruce George May 18 '11 at 6:02

John Conway gives a proof (that no form in 3 or fewer variables is universal) in 2 paragraphs on page 142 of his book, The Sensual Quadratic Form. But the two paragraphs depend on $p$-adic squareclasses and other bits of theory developed elsewhere in the book.

  • $\begingroup$ Thanks! I had heard of the book years ago, it is time I read it. $\endgroup$ – Bruce George May 18 '11 at 6:04

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