Similarity of quadratic and symmetric bilinear forms in characteristic 2

Small zeros of quadratic forms over the algebraic closure of Q

Let $N \geq 2$ be an integer, $F$ a quadratic form in $N$ variables over $\bar{\mathbb Q}$, and $Z \subseteq \bar{\mathbb Q}^N$ an $L$-dimensional subspace, $1 \leq L \leq N$. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space $(Z,F)$. This provides an analogue over $\bar{\mathbb Q}$ of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over $\bar{\mathbb Q}$. We also include some related effective results on orthogonal decomposition and structure of …

Bilinear forms in Weyl sums for modular square roots and applications

Let $q$ be a prime, $P \geq 1$ and let $N_q(P)$ denote the number of rational primes $p \leq P$ that split in the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$. The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for $N_q(P)$ in the range $q^{1/4+\varepsilon} \leq P \leq q$, for any fixed $\varepsilon>0$. This improves upon what is implied by work of Pollack and Benli-Pollack. The second part of this paper is dedicated to proving an estimate for a bilinear form involving Weyl sums for modular square roots (equivalen…

Incidence Matrices of Polarized Projective Spaces

In this paper, we first define a non-degenerate symmetric bilinear form on Fq^4. Then we get an incidence matrix G of Fq^4 by the bilinear form. By its corresponding quadratic form Q, the lines of Fq^4 are classified as isotropic and anisotropic lines. Under this classification, we can get two sub-matrices of G and prove their 2-rank. Key words and phrases: finite field, quadratic form, incidence matrix.

Similarity of quadratic forms over global fields in characteristic 2

Let $ K $ be a global function field of characteristic $ 2 $. For each non-trivial place $ v $ of $ K $, let $ K_{v} $ be the completion of $ K $ at $ v $. We show that if two non-degenerate quadratic forms are similar over every $ K_{v} $, then they are similar over $ K $. This provides an analogue of the version for characteristic not $ 2 $ previously obtained by T.Ono.

拟线性p-形式与全奇异二次型的Vishik等价与相似性

Vishik equivalence and similarity of quasilinear $p$-forms and totally singular quadratic forms

本文将Vishik等价概念推广到拟线性p-形式，证明Vishik等价的p-形式一般不必相似，并找到若干全奇异二次型族使该问题成立。

Symmetric multilinear forms and polarization of polynomials

We study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in $n$ variables. The main tool is combinatorial polarization, and the approach is applicable even when $n!$ is not invertible in the underlying field.

对称双线性形式与正特征中孤立奇点的局部epsilon因子

Symmetric bilinear forms and local epsilon factors of isolated singularities in positive characteristic

研究孤立奇点的对称双线性形式与vanishing cycles复形的局部epsilon因子关系，通过双线性形式的判别式确定符号，改进Milnor公式。

On binary quadratic forms with semigroup property

A quadratic form f is said to have semigroup property if its values at points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all binary integer quadratic forms with semigroup property. If there is an integer bilinear map s such that f(s(x,y))=f(x)f(y) for all vectors x and y from the integer 2-dimensional lattice, then the form f has semigroup property. We give an explicit description of all pairs (f,s) with the property stated above. We do not know any other examples of forms with semigroup property.