Timeline for Integer positive definite quadratic form as a sum of squares
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 14, 2019 at 10:03 | comment | added | Fedor Petrov | ah, great, thank you! | |
| Apr 13, 2019 at 21:38 | comment | added | WKC | 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. | |
| Apr 13, 2019 at 21:35 | comment | added | WKC | 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 $p$. | |
| Apr 13, 2019 at 8:03 | comment | added | Fedor Petrov | Sorry for my ignorance, what does it mean "a quadratic form is represented by the genus of another quadratic form in other number of variables"? | |
| Apr 13, 2019 at 6:13 | comment | added | WKC | Every positive definite quadratic form in $n$ variables is represented by the genus of the quadratic form $x_1^2 + \cdots + x_{n+3}^2$. If $n \leq 5$, that sum of $n + 3$ squares has class number 1 which implies that every positive definite quadratic form in $n$ variables is represented by sum of $n + 3$ squares (again, $n \leq 5$ here). | |
| Apr 13, 2019 at 6:10 | comment | added | Fedor Petrov | ok, and how does it explain that a non-negative definite integer quadratic form in $n$ variables is a sum of $n+3$ squares? | |
| Apr 13, 2019 at 5:58 | comment | added | WKC | I am talking about the quadratic form $x_1^2 + \cdots + x_{n+3}^2$. Its class number is 1 when $n \leq 5$. | |
| Apr 13, 2019 at 4:45 | comment | added | Fedor Petrov | 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? | |
| Apr 13, 2019 at 4:37 | vote | accept | Fedor Petrov | ||
| Apr 13, 2019 at 0:58 | history | answered | WKC | CC BY-SA 4.0 |