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      Questions tagged [quadratic-forms]

      The tag has no usage guidance.

      7
      votes
      3answers
      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
      2answers
      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
      0answers
      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
      0answers
      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
      0answers
      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
      0answers
      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
      0answers
      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
      0answers
      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
      1answer
      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
      0answers
      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
      2answers
      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
      0answers
      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
      0answers
      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
      1answer
      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
      0answers
      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 ...

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