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Results tagged with quadratic-forms
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user 11919
Algebraic and geometric theory of quadratic forms and symmetric bilinear forms, e.g., values attained by quadratic forms, isotropic subspaces, the Witt ring, invariants of quadratic forms, the discriminant and Clifford algebra of a quadratic form, Pfister forms, automorphisms of quadratic forms.
7
votes
A quadratic Diophantine equation
Working in the finite field $\mathbb{F}_p$ and applying a linear change of variables, the equation can be written as
$$a_1x_1^2+a_2x_2^2=1$$ with some nonzero coefficients $a_1,a_2\in\mathbb{F}_p^\ti …
- 105k
2
votes
Accepted
probably Lagrange or Legendre, Pell variant
According to Dickson (History of numbers Vol. 2, Ch. XII, p.376), Göpel (Jour. für Math. 45, 1853, 1-14) proved your conjectures "by use of continued fractions".
Actually Jour. für Math. stands for C …
- 105k
8
votes
Accepted
The quadratic form $x^2+ny^2$ via prime factors
The answer is yes. To see this, consider the ring $R=\mathbb{Z}[\sqrt{-n}]$. If $z=p_1\dots p_k$ is the decomposition of $z$ into rational primes, then by assumption each $p_j$ decomposes in $R$ as $p …
- 105k
7
votes
How many solutions are there to $\sum_{i=1}^3 x_i^2+x_iy_i+y_i^2=k$?
This is a supplement to Noam Elkies' nice answer. The coefficients $s(k)$ can be expressed as
$$ s(k)=27\sum_{d\mid k}\chi(k/d)d^2-9\sum_{d\mid k}\chi(d)d^2, $$
hence the function $\varphi$ is a linea …
- 105k
12
votes
Accepted
how to prove an equation involving sums of Kronecker symbol
The identity can be rewritten as
$$\sum_{\substack{|x|<p\\ 2|x}}\sum_{r|p^2-x^2}\left(\frac{-3}{r}\right)=p+2,$$
because for $x=0$ the inner sum is $1-1+1=1$. Writing $x=2c$, the identity becomes
$$\s …
- 105k
5
votes
Asymptotic formula for sums of four squares?
Stanley Yao Xiao gave a perfect answer, but let me remark that $r_4(n)$ also equals $n$ times the usual singular series (familiar from the circle method) when $4\nmid n$. The difference with five or m …
- 105k
9
votes
Equivalence of binary quadratic forms over $\operatorname{GL}_2(\mathbb{Q}_p)$ or $\operator...
The number of classes of (regular) binary quadratic forms over $\mathbb{Q}_p$ equals $7$ when $p\neq 2$, and it equals $15$ if $p=2$. See Section IV.2.3 of Serre: A course in arithmetic.
The number o …
- 105k
3
votes
About list of discriminants of real quadratic fields with narrow class number 1?
The class number 1 problem was solved only for rather special positive fundamental discriminants. These are cases where the regulator of the underlying real quadratic field is automatically small, so …
- 105k
26
votes
Accepted
When does $axy+byz+czx$ represent all integers?
Here is a proof of the conjecture. I will refer several times to the book Cassels: Rational quadratic forms (Academic Press, 1978).
1. Let $p$ be a prime such that $p\nmid a$. Using the invertible li …
- 105k
6
votes
Accepted
Connection between quadratic forms and ideal class group
There is a concise account in the Appendix of these notes.
- 105k
11
votes
Accepted
Are lattice points in thin spherical shells uniformly distributed?
Yes, they are equidistributed as long as $\delta<11/16$ and $r=R^{-\delta}$ and $R\to\infty$. Without loss of generality, we shall assume that $\delta>-1$ (i.e. $r<R$).
To see this, let $\mathcal{F}\s …
- 105k
5
votes
Accepted
Duke and Schulze-Pillot condition for equidistribution
I recommend that you study the paper by Duke and Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids (Inventiones, 1990) …
- 105k
3
votes
Accepted
Primitive representation of integers by some form on the genus of a quadratic form
The quoted text was written by me. As I corrected myself recently in a comment below the original post: I secretly assumed that $n$ was coprime to $\det(a_{ij})$. In that case, I believe that modulo $ …
- 105k
3
votes
Accepted
On certain solutions of a quadratic form equation
Your ultimate question was answered by Gauss: $O_f^- \cap \operatorname{GL}_2(\mathbb{Z})$ is nonempty if and only if the class of $f$ is ambiguous (i.e. its square is the trivial class).
Indeed, $f( …
- 105k
21
votes
Accepted
two's and three's survive in gcd of Lagrange
The claim is certainly true for $n$ sufficiently large, and "sufficiently large" could be specified explicitly with more care.
We follow the suggestion of Fedor Petrov, and rely on the results of Br …
- 105k