# Questions tagged [algebra]

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### Power of non-implicationally-complete Frege systems and Boolean equational calculus

We know that Frege systems are required to be implicationally complete -- namely, if a set of formulas $B_1,B_2,\cdots,B_t$ imply formula $C$, then this implication can be proven in the system. I'm ...
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### Product types: algebraic structure for modeling product types with commutative and associative product operation

Is there a known algebraic structure over set of Types (however they are defined) which is equipped with: commutative and associative product operation for building product types from simpler types, ...
130 views

### Complexity of solving a higher-order degree polynomial equation? P-problem or NP-problem or neither?

I am a mathematician and I am very new to theoretical computer science. The definition of P/NP problem I found in wiki is that: P is the set of decision problems solvable in polynomial time by a ...
122 views

### Can the initial algebra of a 2-variable polynomial functor be computed on the diagonal?

Given a polynomial functor $F$, its initial algebra is denoted by $\mu X.F(X)$. Now, if $F$ is a 2-variable polynomial functor, $Y \mapsto \mu X.F(X,Y)$ turns out to be functorial and we can, again, (...
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1 vote
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### Rational power series over $\mathbb N \cup \{\infty\}$, rationality of singular part

Let $\Sigma$ be a finite alphabet, and consider the formel power series over $\Sigma$ considered as non-commuting variables with coefficients in the semiring $\mathcal N := \mathbb N \cup \{\infty\}$ ...
161 views

### Example of monoid $M$ such that $\operatorname{RAT}(M) \not\subseteq \operatorname{REC}(M)$

Let $M$ be a monoid, the family of rational sets $\operatorname{RAT}(M)$ is defined as the smallest set containing the finite subsets, and closed under union, concatentaion and the star operation. The ...
242 views

508 views

### 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 ...
111 views

### 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. https://math....
2k views

### 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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
386 views

### 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 ...
229 views

### 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 ...
1 vote
68 views

### 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 ...
649 views

### 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 ...
1k views

### 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 ...
144 views

### 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 $1 \leq m \leq k$? ...