# Questions tagged [quadratic-forms]

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**7**

votes

**3**answers

226 views

### Integer positive definite quadratic form as a sum of squares

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 $Q$...

**2**

votes

**2**answers

156 views

### Solving diagonal simultaneous quadratic equations

A problem I am trying to solve has led to me to the following system of equations:
$$A(x^2) + Bx + c = 0$$
Where $A$ and $B$ are known matrices, $c$ is a known vector, $x$ is the vector of unknowns ...

**1**

vote

**0**answers

83 views

### Norm quadrics and their motives

Let $k$ be a field of characteristic $\neq 2$ and $\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle$ a Pfister form over $k$. Denote by $Q_{\underline{a}}=Q_{a_{1},\cdots,a_{n}}$ the projective ...

**6**

votes

**0**answers

102 views

### Is it possible for the Witt group of a scheme to have non-trivial odd torsion?

Let $F$ be a field of characteristic not $2$. A well-known theorem of Pfister asserts that the torsion of $W(F)$, the Witt group of $F$, is $2$-primary.
Baeza [B, V.6.3] extended this result to Witt ...

**0**

votes

**0**answers

102 views

### A conjecture stronger than Gauss' triangular number theorem

A subset $A$ of $\mathbb N=\{0,1,2,\ldots\}$ is called an asymptotic additive basis of order $3$ if any sufficiently large integer can be written as the sum of three elements of $A$.
I have ...

**1**

vote

**0**answers

87 views

### Nonnegative integers written as $x(3x+1)+y(3y-1)+z(3z+2)+w(3w-2)$ with $x,y,z,w\in\{0,1,2,\ldots\}$

Recently, I formulated the following conjecture which is somewhat similar to Lagrange's four-square theorem.
Conjecture. Each $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $x(3x+1)+y(3y-1)+z(3z+...

**3**

votes

**0**answers

144 views

### On sums of three squares

Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, any $n\in\mathbb N$ with $n\equiv1,2\pmod4$ can be written as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$. Clearly, ...

**5**

votes

**0**answers

201 views

### Sums of two integer squares in arithmetic progressions

Is there an explicit formula in the literature for the number of representations of a positive integer $n$ as a sum of two integer squares, the second of which is divisible by $5$? So this means to ...

**0**

votes

**1**answer

110 views

### A “nice” (but non-definite) quadratic programme

For integers $n\geq k>0$, let $f$ be the following quadratic form:
$$f(x_1,\ldots,x_n)=\sum_{i=1}^n\sum_{j=0}^{k-1}x_ix_{i+j\bmod n}.$$
Is it true that the minimum of $f$ over the unit simplex is ...

**2**

votes

**0**answers

80 views

### On universal sums $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ over $\mathbb N$

Let $a,b,c,d,e,f$ be integers with $a\ge c\ge e>0$, $b>-a$ and $a\equiv b\pmod2$, $d>-c$ and $c\equiv d\pmod 2$, $f>-e$ and $e\equiv f\pmod2$. If each $n\in\mathbb N=\{0,1,2,\ldots\}$ can ...

**2**

votes

**2**answers

88 views

### Correlation between the first and a random position of an ergodic bit sequence

Edit: Since the geometric approach did not work, I try now another approach: phrasing the problem as a quadratic programme.
Probabilistic version.
Let $x=(x_1,x_2, \ldots) $ be an ergodic random ...

**1**

vote

**0**answers

37 views

### How explicitly write a projective transformation between the conics over the univariate function field?

Consider the quadratic forms
$$
Q_1 = x^2 + y^2 - (t^2+1)z^2,\qquad Q_2 = x^2 + y^2 - z^2
$$
over the rational function field $\mathbb{F}_p(t)$, where $p > 2$ is a prime such that $t^2 + 1$ is ...

**0**

votes

**0**answers

53 views

### Quadrics over the univariate function field with discriminant of minimal degree

Consider a non-degenerate quadric $Q(x,y,z) \subset \mathrm{P}^2$ over the univariate function field $\mathbb{F}_p(t)$, where $\mathbb{F}_p$ is a prime finite field, $p > 2$. For simplicity assume ...

**4**

votes

**1**answer

91 views

### On the orthogonal group of a lattice on a quadratic space over dyadic local field

Let $F$ be a local field with valuation ring $R$. $V$ is a n dimensional non-singular quadratic space over $F$ with bilinear form $B$ and quadratic map $Q$.
As usual, $O(V)$ denotes the orthogonal ...

**1**

vote

**0**answers

83 views

### Concrete Hanson-Wright inequality?

I'm working on a paper that requires bounding
$$\Pr\left[|\vec x^\top Q \vec y| >= t\right]$$ where $Q$ is a matrix (happens to be symmetric) and $\vec x,\vec y$ are iid real mean-zero subgaussian ...