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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.
12
votes
Accepted
How to prove this problem about ternary quadratic form?
Let $G(n)$ denote the weighted number of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes of positive definite quadratic forms
$$[a,b,c]:=ax^2+bxy+cy^2$$
with $2\mid b$ and $b^2-4ac=-4n$, where the cla …
- 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
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
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 …
2
votes
Correspondence between binary quadratic representations and proper ideals of quadratic numbe...
We shall give an explicit correspondence between $R(d,n)$ and $I(d,n)$ based on the explicit correspondence between representatives $Q$ of proper equivalence classes of quadratic forms of discriminant …
- 105k
14
votes
Accepted
Set of quadratic forms that represents all primes
Every prime $p$ is represented by at least one of the following quadratic forms: $x^2+y^2$, $x^2+3y^2$, $3x^2-y^2$:
if $p=2$ or $p\equiv 1\pmod{4}$, then $p$ is represented by $x^2+y^2$;
if $p=3$ or …
- 105k
4
votes
Accepted
On quadratic forms in four variables
It follows from Deligne's bound for the Hecke eigenvalues of weight $2$ holomorphic cusp forms (which is really Eichler's theorem in this special case) that the error term is $O_{F,\varepsilon}(m^{\fr …
- 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
$$\s …
- 105k
20
votes
Accepted
Which quaternary quadratic form represents $n$ the greatest number of times?
Theorem. Let $Q(x_1,\dots,x_k)$ be a positive definite integral quadratic form in $k\geq 2$ variables. Then the number of integral representations $Q(x_1,\dots,x_k)=n$ satisfies
$$r_Q(n)\ll_{k,\epsil …
- 105k
20
votes
Many representations as a sum of three squares
Let me restrict to the number of primitive representations
$$r_3^\ast(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n\ \text{and}\ \gcd(a,b,c)=1 \}\right|.$$
Note that $r_3(n)$ can be easily e …
- 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
4
votes
Accepted
Strong Approximation for solutions to quadratic Diophantine equations
The conjecture follows from Theorem 2.1 in Hsia-Jöchner: Almost strong approximations for definite quadratic spaces, Inventiones 129 (1997), 471-487. The paper is available here.
The details of this …
- 105k
1
vote
Accepted
Elementary description to count of perfect squares - I
As hinted in Alex Kruckman's comment, it is easier to work with
$$N'(a) = \left|\{x\in \{1,\dots,a-1\} : \sqrt{x(a-x)}\in \mathbb{Z}\}\right|,$$
which is the same as $N(a)$ up to a factor of $2$.
It i …
- 105k
20
votes
Accepted
A diophantine equation in $\mathbb{N}$
This is an elaboration of Emil Jeřábek's important comment, and contains no original contribution. The OP's problem was examined in depth by Borwein-Choi (1999), and their article is available for fre …
- 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