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Results tagged with finite-fields
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user 31356
A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.
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Degree of a field extension with a rational solution
Let $S$ be a system of polynomial equations over $\mathbb{F}_q$.
Assume that $S$ has a solution in $\overline{\mathbb{F}_q}$.
Denote by $k$ the minimal number such that $S$ has $\mathbb{F}_{q^k}$- …
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vote
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88
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Calculation of cardinality of Jacobians
The problem of calculation the number of rational points on curves over finite fields is $\#P$-complete - "Counting curves and their projections".
Is it true for calculation of number of rational p …
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Zeta function of $\Delta[\text{det},m]$
In Geometric complexity theory the following variety $\Delta[\text{det},m]$ is crucial.
Let $X=(x_1,\ldots,x_r)$ be a tuple of $r=m^2$ variables, so that $X$ can be thought of as an $m\times m$ varia …
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5
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Schwartz-Zippel lemma for an algebraic variety
Let $X $ be a smooth affine subvariety of $(\overline{\mathbb{F}_q})^n$ defined by a prime ideal $I$. Let $f$ $\in \mathbb{F}_q[x_1,\ldots,x_n]$ be a polynomial such that $f \notin I$.
Let $r_1, \ld …
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Geometric complexity theory for finite fields
Geometric complexity theory (GCT) is an approach via algebraic geometry and representation theory towards the P vs. NP problems and related problems Ketan D. Mulmuley.
More precisely, the idea is to …
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answers
194
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Number of common solutions of polynomial system
Let $ \mathbb{F}_p$ be a finite field and $\{f_j\}_{j=1}^{j=r} \subseteq \mathbb{F}_p[X_1,...,X_n]$ be a set of polynomials.
Let consider the system of equations:
$f_j(x_1,...,x_n)=0$ for $j = 1,.. …
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4
votes
2
answers
541
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Irreducible algebraic sets via irreducible polynomials
There are many results about irreducible polynomials over finite fields:
we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, …
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vote
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answers
152
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Efficient deterministic algorithms of factorizing
My question is about efficient deterministic algorithms of factorizing polynomials of degree $n$ over $\mathbb{F}_q$.
Are there such algorithms that use poly$(n, \log q)$ bit operations?
I know tha …
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3
votes
0
answers
386
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Calculation of Cartier-Manin matrix
Let $\mathbb{F}_q$ be a finite field of characteristic $p$ and let $C$ be a plane projective nonsingular curve over $\mathbb{F}_q$ ,
with function field $K = \mathbb{F}_q(C)$. Let $K^p$ denote the sub …
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Equations of elliptic curves
Curve $E$ with equation $y^2 = x^3 + 1$ has $p+1$ rational points over $\mathbb{F}_p$ when $p = 5$ (mod $6$). Let $q = p^n$. $|E(\mathbb{F}_{q^2})| = q^2 \pm 2q +1$.
Let $\zeta_6$ is generator of $\m …
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1
vote
2
answers
470
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Equations of elliptic curves
First part of question I have asked on mathoverflow already: https://math.stackexchange.com/questions/467088/explict-form-of-the-equation-of-elliptic-curve
1) Let $E(\mathbb{F}_{q^2})$ is elliptic cu …
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13
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answer
1k
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An efficient isomorphism between finite fields
Let $p$ be a prime number. Let $f$ and $g$ be irreducible polynomials over $\mathbb{F}_p$, both of degree $n$. We know that factor-rings $\mathbb{F}_p[x]/(f)$ and $\mathbb{F}_p[x]/(g)$ are isomorphic …
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