When is a given polynomial a square of another polynomial?

I meet a problem in which I hope to show a special polynomial is not a square of another polynomial. More precisely, let’s consider the polynomial f(x):=1−x+2bxn−2bxn+1−b2x2n−1+2b2x2n−b2x2n+1−2bx3n−1+2bx3n−x4n−1+x4n∈k[x], where k is a field of characteristic p>0, n>2 is an integer, and b∈k with b≠0,1,−1. Indeed, in the context I meet, the field k is just the … Read more

Chevalley-Warning for finite rings: the degree of a non-polynomial

\def\F{\mathbb F} \def\Z{\mathbb Z} One reason that Chevalley-Warning theorem is that amazingly useful is the fact that for a finite field \F, any function from \F^n to \F is a polynomial. This fails to hold in general for finite rings; say, it is easy to see that if R is a finite (commutative) ring but … Read more

Polynomial equations in many variables have solutions (Lang 1952 paper)

I am trying to understand the proof of the following result: Suppose F is a function field in k variables over an algebraically closed field. Let f1,…,fr∈F[x1,…,xn] be polynomials without constant term of degrees d1,…,dr, respectively. Assume that n>dk1+…+dkr. Then f1,…,fr have a non-trivial common zero. If F is instead a function field in k … Read more

When spreading out a scheme, does the choice of max ideal matter?

I’m looking at Serre’s paper How to use Finite Fields for Problems Concerning Infinite Fields. Specifically I’m trying to use the techniques in the proof of Theorem 1.2 to write out the details of the proof of Theorem 3.1, but I’ll try to ask a more general question. Here’s how I currently understand spreading out … Read more

Monoid cohomology of N\mathbb{N} for a linear algebraic group

Let k be a finite field and kE:=k((X)) denote the field of Laurent series over k. We define a Frobenius endomorphism on kE via f(X)↦f(Xp). We choose a lift φ:ksepE→ksepE on the seperable closure of kE denoted by ksepE=(kE)sep (eg. if k=Fp, we can choose φ:=(⋅)p). Now let G be a linear algebraic group over … Read more

Counting the image of a map of varieties using the trace formula

Suppose f:X→Y is a finite map of varieties over a finite field Fq. Is there an etale constructible Qℓ sheaf F on Y which counts the number of rational points of the form f(x) for x itself rational (as an application of the trace formula)? If f is closed, we can just use the pushforward. … Read more

δ\delta-equidistributed polynomials over finite fields

I’m trying to show that a polynomial over finite (prime) field is “close enough” to being equidistributed over its range. A polynomial p(⋅) from Fn to F is δ-equidistributed is ∀c∈F,|{x∈Fn:p(x)=c}|=(1±δ)|F|n−1. I found a paper by Green and Tao that deals with it. They define the notion of rank of a polynomial: rankd−1(P) is the … Read more

Vandermonde matrices and general position

I was wondering if it is known whether a Vandermonde matrix over a sufficiently large finite field is in general position with respect to intersections of subspaces spanned by subsets of columns, i.e. whether the dimension of such intersections is as small as that of a randomly chosen matrix over a large field. To be … Read more

Trivializing principal bundles on a curve over a finite field

This is related to my question Adelic description of moduli of G-bundles on a curve. Let X be a smooth, projective, and geometrically connected curve over a finite field Fq and G a smooth connected affine algebraic group over Fq. Under what conditions on G does any G-bundle on X trivialize over a Zariski open … Read more

probability of having linearly independent sparse vectors over finite fields

Suppose that there are $i< N$ linearly independent $N$-dimensional vectors $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_i$ over a finite field $\mathbb{F}_q$, whose elements are denoted as $\{0,1,2,\ldots,q-1\}$. Each vector consists of exactly $m$ nonzero elements uniformly randomly chosen from $\{1,2,\ldots,q-1\}$, $m\ll N$. The nonzero coordinates of each vector are randomly located. Now the question is, given a new such random … Read more