# Tagged Questions

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### Chomsky Schützenberger enumeration theorem

In many textbooks the Chomsky-Schützenberger enumeration theorem is stated as that the characteristic formal power series of a language is $\mathbb N$-algebraic, if the grammar is unambigious. In some ...
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### Algebraic dependence of roots of irreducibles over a finite field

I asked this question in Math SE too, but I have since modified it to make it more suited here. Also, in hindsight, the question itself was more algorithmic and was a better fit here. http://math....
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### On the realisation of monoids as syntactic monoids of languages

Let $L \subseteq X^{\ast}$ be some language, then we define the syntactic congruence as $$u \sim v :\Leftrightarrow \forall x, y\in X^{\ast} : xuy \in L \leftrightarrow xvy \in L$$ and the quotient ...
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### Counting points on curves

It is known (see "Counting curves and their projections" (free version) by von zur Gathen, Karpinski, and Shparlinski) that the problem of finding the number of $\mathbb{F}_q$-rational points on a ...
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### What's the relationship between “free theorems” and “free objects”

What's the relationship between free theorems and free objects from algebra. They seem quite similar. I'm wondering if there's an underlying principle here.
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### Finding of dimension of algebraic varieties

I have found that the problem of finding of dimension of algebraic varieties over $\mathbb{C}$ is $NP$-complete (https://pdfs.semanticscholar.org/a947/463a29ee512b89823176f6e8c9f9b2bb1a5e.pdf). Are ...
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### Computing $a^e \mod p^n$ Efficiently

It is well known that we can compute: $$a^e \mod m$$ in $O(\log e \log ^2 m)$ bit operations (assuming multiplication $nm$ in $O(\log n \log m)$ time) via exponentiation by squaring. I am wondering ...
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### Algebra and algebraic data types

Which of the well-known structures of modern algebra (monoids, groups, rings etc) can be expressed as algebraic data types (ADTs)? Presumably a free monoid can be considered to be isomorphic to the ...
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### Is algebraic dependency decidable?

A set of numbers $S=\{x_1,...,x_n\}$ is said to be algebraically dependent if there exists a (multivariate) polynomial $p$ with coefficients in $\mathbb Q$ whose roots contain $x_1,...,x_n$ (or a ...
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### Complexity of eigenvaue problem

Many matrix diagonalization algorithms have time complexity $\mathcal{O}(n^3)$ where $n$ is the number of columns/raws (consider only square matrices). What is the best time lower bound it is known? ...
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### The curve used in Parvaresh-Vardy decoding

Consider the Parvaresh-Vardy list decoder. As I understand it, the idea is to decide on a curve over an extension field of the form $(f,f^h mod E, f^{h^2} mod E,\dots)$ and then evaluate each of ...
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### Complexity of computing the order of a permutation group

Given two permutations $g$ and $h$ over $n$ elements (i.e., members of $S_n$), what is the complexity of computing the order of the subgroup generated by $g,h$? Or just of deciding whether the ...
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### List of number theoretic or algebraic problems in various complexity classes

I am looking for a list about the known or unknown complexity of various number theoretic /algebraic problems. For example, GCD in $NC^1$ is open, factoring in $P$ is open, computing sheaf ...
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### Solving a system of sums-of-powers polynomials

What is the complexity of calculating the values of the integers $x_i$, where $0 \leq x_1 < x_2 < \dots < x_k < n$, given only the values $s_m = \sum_{i=1}^k x_i^m$? for $1 \leq m \leq k$? ...
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### Algebra oriented branch of theoretical computer science

I have a very strong base in algebra, namely commutative algebra, homological algebra, field theory, category theory, and I am currently learning algebraic geometry. I am a math major with an ...
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### Computability of infinite-dimensional vector space

So there is a talk about infinite-dimensional vector space being computable. But then I find it hard to understand. Apparently, dimension is infinite, so how would the operations of the space be ...
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### What is necessary and/or sufficient requirement for a subring of a field to be computable? [closed]

As title asks, what is necessary and/or sufficient requirement for a subring of a field to be a computable ring? Conditions on either field or subring are fine.
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### Decoding of Gabidulin codes

Consider the space of matrices in $\mathbb{F}_q^{n \times m}$ where $\mathbb{F}_q$ is the finite field with $q$ elements. We can define a metric on this space, given by $d(A,B) := rank(A-B)$, called ...
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### Restoring symmetry in certain combinatorial bijections?

I'm interested in two 'natural bijections' that involve labeled forests and Young tableaux. Let me give the definition for labeled forests. By this, we mean a pair $\cal{F} = (F,f)$ where $F$ is an $n$...
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### Classifying noetherian simple groups by order type?

A (possibly infinite) group $G$ is noetherian if it satisfies the following equivalent conditions: (1) every subgroup of $G$ is finitely generated, (2) there is no infinite strict ascending chain of ...
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### Spectrum of a variety: a possible connection btw ordinals and structures?

Consider a variety of algebras $\mathbb{V} = \mathbb{V}(\sigma,\tau)$ which consists of the set of algebras defined over a fixed signature $\sigma$ and satisfying a set of identities $\tau$. We may ...
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### Possible generalizations of associativity?

The well-known notion of associativity in algebra leads to structures with interesting properties, such as groups or semigroups. According to a paper by John Rhodes, some researchers in algebra and ...
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### Composition series and isogeny

I'm not sure this question is appropriate for this site, but it might have some connections with computational algebra. Consider a fixed "category" $\sf{Cat}$ (in the sense of category theory, but ...
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### Extending the notion of independence

Background I was looking for a formulation of 'free sets' and 'independent sets' from linear algebra that would extend to groups. This question was considered here but I couldn't find a satisfactory ...
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### Extending semigroup theory?

In an earlier question I proposed a definition of associativity for ternary relations generalizing the usual notion for composition laws. I'm still not sure whether this definition makes sense, but if ...
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### Where does randomness help when deciding algebraic geometry over $\mathbb{C}$?

If we have a single straight line program expressing a multivariate polynomial equation with integer coefficients, the Schwartz-Zippel lemma gives a simple randomized algorithm for deciding whether ...
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### The polynomial languages and ordered syntactic monoids

A polynomial language is a languge which could be represented as the finite union of languages of the form: $$A_0^* a_1 A_1^* a_2 \cdots a_k A_k^* \quad a_i \in X, A_i \subseteq X$$ Such an ...
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### Why is it difficult to develop a subexponential functoral approach to discrete log

Call a discrete logarithm algorithm functoral if the commutative algebraic object that the algorithm acts on can be changed with another commutative object in the same category and the algorithm does ...
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### Factoring low-degree polynomials

What is the fastest algorithm known for factoring polynomials with $n$ variables and total degree $\leq d$? Here, $n$ is growing and $d$ is fixed. Most work seem to consider the case when $d$ is ...
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### Equivalence relations on strongly regular graphs with same parameters

Let $\mathcal{S}$ be the set of all strongly regular graphs with parameter $(n, k, \lambda, \mu)$. Are there any (interesting) equivalence relations defined on this set? My motivation is to approach ...
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### Algebraic (or numeric) invariants of complexity classes

I hope this question isn't too naive for this site. In mathematics (topology, geometry, algebra) it is common for one to distinguish between two objects by coming up with an algebraic or numerical ...
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### count number of i such that ( (a*i+b) mod p) mod k == l

How to determine the number of $i$'s as fast as possible such that $$1\le i \le L and ((ai+b)\mod p) \mod k = l$$ where $p$ is a prime number, $1\lt a, b\lt p-1$, and $l \lt k \lt L \lt p$. This ...
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### How to go about proving the basic operators in relational algebra are independent of each other?

The five basic operator select, project, cross, union and diff in relational algebra are independent of each other. I'm trying to formally prove this statement but can only progress for cross product ...
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### Is there a theory that combines category theory/abstract algebra and computational complexity?

Category theory and abstract algebra deal with the way functions can be combined with other functions. Complexity theory deals with how hard a function is to compute. It's weird to me that I haven't ...
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### What's the bias of random polynomials with low degree over GF(2)?

I have a question concerning low-degree polynomials and probability: What is the (assyptotic behavior of the) probability that a random* polynomial, $p$, over GF(2), with degree $\le d$ and n ...
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### Sufficient conditions to guarantee unique fixpoint (not unique least/greatest fixpoint) for monotone functions on complete lattice

Tarski's fixpoint theorem states, that the fixpoints of a monotone operator on a complete lattice is a complete lattice. By consequence, we have a unique greatest fixpoint and unique least fixpoint ...