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Informally, an algorithm is a set of explicit instructions used to solve a problem (e.g. Euclid's algorithm for computing the greatest common divisor of two integers). For more specific questions on algorithms, this tag may be used in conjunction with the approximation-algorithms, algorithmic-randomness and algorithmic-topology tags.

33 votes
3 answers
6k views

Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$...

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$, decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective, respectively, injective? -- And …
4 votes

Permutation search problems with no known $o(n!)$ algorithms

If you are also interested in problems of that type where $n = \infty$: Given a mapping $f: \mathbb{N} \rightarrow \mathbb{N}$ from the natural numbers to themselves, it is often a notoriously hard pr …
Stefan Kohl's user avatar
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12 votes

Is there a way of canonically labelling permutation groups?

A quick way to obtain canonical conjugates of permutation groups would of course be nice, but hoping for that may be a bit too optimistic. Rather than trying to go that route, in your situation I woul …
Stefan Kohl's user avatar
  • 19.6k
6 votes

Algorithm for solving systems of linear Diophantine inequalities

GAP provides a function NullspaceIntMat which solves systems of linear diophantine equations. The documentation says: 25.1-2 SolutionIntMat * SolutionIntMat( mat, vec ) ───────────────────────────── …
Stefan Kohl's user avatar
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23 votes
5 answers
1k views

Securing privacy of "who communicates with whom" under Orwell-like conditions

Assume that there is a big and powerful country with an information-greedy secret service which has backdoors to all internet nodes throughout the world which permit him to observe all exchanged data …
2 votes

Does a classification of simultaneous conjugacy classes in a product of symmetric groups exist?

For the sake of simplicity, consider only the case $d=2$. In this case, two pairs $(a,b), (a,c) \in {\rm S}_n^2$ lie in the same orbit if and only if there is a permutation $\pi$ in the centralizer of …
Stefan Kohl's user avatar
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14 votes

Is there an algorithm to solve quadratic Diophantine equations?

Let me just add that for solving quadratic diophantine equations in 2 variables, i.e. equations of the form $$ ax^2 + bxy + cy^2 + dx + ey + f = 0, \ \ a, b, c, d, e, f \in \mathbb{Z}, $$ there is a …
Stefan Kohl's user avatar
  • 19.6k
4 votes

How to quickly determine whether a given natural number is a power of another natural number?

The computer algebra system GAP performs this test and determines a smallest root $a$ of a given integer $n$ quite efficiently. The following is copied directly from its source code (file gap4r6/lib/i …
Stefan Kohl's user avatar
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2 votes
Accepted

Find root in finite field

Pasting the title of your question into Google gives references like http://www.math.leidenuniv.nl/~astolk/monday/notes/stolk-roots.pdf and http://www.ma.utexas.edu/users/voloch/Preprints/roots.pdf -- …
Stefan Kohl's user avatar
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