All Questions
7 questions with no upvoted or accepted answers
10
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436
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Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?
$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons.
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- 24.9k
7
votes
0
answers
276
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Cyclic shift acting on finite Grassmannian
The (twisted) cyclic shift $(v_1,v_2,\ldots,v_n) \mapsto (v_2,v_3,\ldots,v_n,(-1)^{k-1}v_1)$ acting on the Grassmannian $\mathrm{Gr}(\mathbb{C};k,n)$ of $k$-planes in $\mathbb{C}^n$ is an important ...
- 24.2k
6
votes
0
answers
314
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Number of square-free polynomials over a finite field - a combinatorial interpretation?
Cross-posted from MSE. The question has remained unanswered for six years but I still like it!
One can show using zeta functions that the number of (monic)square-free polynomials of degree $n$ over a ...
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3
votes
0
answers
293
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Approximate versions of Segre's Theorem
Consider projective $2$-space over a finite field of odd prime characteristic $p$. We say a set of points, $A$, in this space is an arc if any line meets it in at most two points. We say that an arc ...
- 13k
1
vote
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75
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Symmetric matrices of hyperbolic and elliptic type with certain kind of trace zero
I have been working on a problem related to determinantal varieties in symmetric matrices. I am stuck at the following point and would like to get some reference/help for the following question.
Let $\...
- 179
1
vote
0
answers
207
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Polynomial existence over finite field
Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$.
Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$).
Denote $e_i=(0,\dots,0,\...
- 13.9k
0
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A question on bounding the size of the polynomial
Suppose we are given the following n polynomials in $\bar{\mathbb{F}}_2[x_1,...,x_n]$:
$f_1 = x_1 + x_n^2$
$f_2 = x_2 + x_1^2$
$\cdot$
$\cdot$
$f_{n-3} = x_{n-3} + x_{n-4}^2$
$f_{n-2} = x_{n-2} + x_{n-...
- 341