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.

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### Computing coefficients of theta functions associated to quadratic forms

If we take an integral positive definite quadratic form $Q$ and set $\Theta_Q(z) = \sum_{k\geq 0}R_Q(k)e^{2\pi ikz}$, what are the most efficient algorithms to compute the $R_Q(k)$? I am aware e.g. of ...
1 vote
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### Existence of solution for a system of quadratic diophantine equations / symmetric quadratic froms

I am interested in solving, or even just deciding the existence of a solution, for a system of quadratic diophantine equations. Let $p$ be a prime congruent to 1 modulo 8, so $p =17$ is the first ...
157 views

### Product of Hermitian forms over a group ring

Let $G$ be a group and $k=Z[G]$ the corresponding group ring equipped with the standard involution. Let $(P,p)$ and $(Q,q)$ be two $\epsilon$-quadratic forms, $\epsilon = \pm1$, where $P$ and $Q$ are ...
175 views

### Integers representable as binary quadratic forms

It is known that odd prime $p$ can be represented as $p=x^2+y^2$ if and only if $p \equiv 1$ mod $4$, represented as $p=x^2+2y^2$ if and only if $p \equiv 1$ or $3$ mod $8$, represented as $p=x^2+3y^2$...
1 vote
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### Can $12n+5$ be written as $2x^2+5y^2+9z^2+xyz$ with $x,y,z$ nonnegative integers?

Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, for any $n\in\mathbb N$ we can write $4n+1$ as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$. Motivated by this, here ...
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### Determinants of minors occurring 'within' determinant of full matrix

$A= (a_{ij})$ is an $n\times n$ symmetric positive matrix. It induces a quadratic form $f(x):= x^tAx$ on $\mathbb{R}^n$. $D_m$ denotes the determinant of the top left $m\times m$ submatrix of $A$ (or ...
154 views

### Proving an optimization problem from continuous input to binary is optimal

Suppose we have a function $f(x,y,z)$ where the inputs are uniform from 0 to 1. The output is either $+1$ or $-1$. And there is a partial symmetry $f(x,y,z) = f(z,y,x)$. Tell me what the minimum of ...
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### Simultaneous diagonalization of two rational forms

It is known that any two real quadratic forms are equivalent iff they have the same signature. If we consider rational quadratic forms, they are $\mathbb{Q}$-equivalent iff the have the same signature,...
276 views

### Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?

I seek for very sparse representations of positive integers. Let $$\mathbb N=\{0,1,2,\ldots\}\ \ \ \text{and}\ \ \ \mathbb Z^+=\{1,2,3,\ldots\}.$$ Recall that a polynomial $P(x,y)$ is integer-valued ...
280 views

### When are rings of the form $K[x_1,...,x_n]/(Q)$ principal ideal domains when $Q$ is quadratic?

By a result of Klein-Nagata rings of the form $A_Q=K[x_1,...,x_n]/(Q)$ are factorial when $K$ is a field, $n \geq 5$ and $Q$ is a non-degenerate quadratic form. Question 1: When is $A_Q$ a principal ...
206 views

### Computing the genus of certain ternary indefinite lattices

For $k$ a positive integer, let us consider the rank 3 ternary indefinite lattice $L=L_k$ with quadratic form $$6kx^2-2(y^2+yz+z^2).$$ Its discriminant group has length $2$. Question. Is this lattice ...
256 views

### Classification of vector spaces with a quadratic form and an order n automorphism

Introductory general nonsense (for motivation: feel free to skip): Let $G$ be a finite group and $k$ be a field of characteristic $0$. Consider the set $\mathcal{S}$ of isomorphism classes of finite ...
1 vote