All Questions
Tagged with algebra cc.complexity-theory
23 questions
3
votes
0
answers
57
views
Complexity of minimizing the index of a subgroup of the free group
Let $\Sigma$ be a finite alphabet and $G$ the free group generated by $\Sigma$. Let $W$ be a finite subset of $G$. (Represented as a list of formal expressions of the form $a_1^{\pm 1}\ldots a_n^{\pm ...
- 2,181
4
votes
1
answer
86
views
Reference request: finite field computation over the Word-RAM model
Let $q = p^\ell$ be a positive integer power of a prime $p$, of size $q = \text{poly}(n)$.
Over the Word-RAM model (with words of size $O(\log n)$), how quickly can we perform addition and ...
- 696
0
votes
1
answer
312
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 ...
0
votes
0
answers
75
views
Polynomial GCD exact complexity in terms of degree and number of variables
https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor#Proof_that_GCD_exists_for_multivariate_polynomials states multivariate polynomial GCDs may be defined over $\mathbb F_p[x_1,x_2,\dots,...
- 13.3k
2
votes
1
answer
201
views
Complexity of finding approximate solutions for systems of polynomial equations
Consider the following problem:
Input: $(p_1,...,p_n, \epsilon)$ where each $p_i$ is a polynomial in $m$ variables with integer coefficients and $\epsilon>0$.
Output: If there is $(r_1,...,r_m) \in ...
- 23
7
votes
1
answer
363
views
Technical lemma about curves used in original proof of PCP theorem
I am reading the proof from here and found a technical lemma that seems to be incorrect (its proof is short and very vague). I know this is rather specific and the context is problematic, but I couldn'...
- 153
7
votes
0
answers
68
views
Complexity of deciding whether subspaces of Z_2^n cover every point 3*x times
When studying the complexity of checking identities in certain finite algebras, I came across the following decision problem:
Input: A positive integer $n \in N$ and a set of affine subspaces $H_1,...
- 71
4
votes
1
answer
391
views
If a root||nonce Proof-of-Work certificate is prime, can it be used in any other interesting proofs?
Because Bitcoin and many other cryptocurrency mining certificates are "rare" in that their respective hash is less than a very small number, can we leverage their rarity in probabilistic proofs of ...
- 1,145
3
votes
1
answer
173
views
Connection between algebraic logic and computational complexity of logics?
I'm learning a bit about algebraic logic and I was wondering how knowing the algebraic semantics of a given logic might help the study of the logic itself from a computational point of view.
In ...
- 1,698
9
votes
0
answers
304
views
What does a private coin $\mathsf{IP}$ protocol for Hilbert's Nullstellensatz look like?
$\mathsf{GNI}$ Private Coin
In [GMW85], the authors provided the famous interactive proof $\mathsf{IP}$ of Graph Non Isomorphism $\mathsf{GNI}$.
The $\mathsf{GNI}$ protocol entails a verifier ...
- 1,145
2
votes
1
answer
102
views
the number of rational points of a curve modulo 2
Consider the language $L=\{f, q\}$ - the number of solutions of equations $f(x,y)=0$ in $\mathbb{F}_q^2$ is equal to zero modulo $2$, where $q = 2^m$. Does $L$ belong to $P$? to $NP$? ($f$ is written ...
- 2,023
1
vote
0
answers
90
views
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 ...
- 2,023
5
votes
0
answers
81
views
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 ...
- 2,023
1
vote
0
answers
119
views
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 ...
- 111
10
votes
2
answers
684
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 ...
- 10.6k
12
votes
2
answers
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 ...
- 13.3k
8
votes
1
answer
310
views
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 ...
- 7,030
8
votes
2
answers
539
views
Bivariate low-degree polynomial testing of Polishchuk-Spielman
In the seminal paper of Polishchuk and Spielman where they give a construction of nearly linear sized $PCP$ for an $NP$ problem, one of the key ingredients is a low-degree test for bivariate ...
- 81
9
votes
3
answers
607
views
Find the remainder of a large fixed polynomial when divided by a small unknown polynomial
Assume we operate in a finite field. We are given a large fixed polynomial p(x) (of, say, degree 1000) over this field. This polynomial is known beforehand and we are allowed to do computation using a ...
- 91
8
votes
1
answer
272
views
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 ...
- 1,328
18
votes
2
answers
2k
views
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 ...
- 1,073
41
votes
12
answers
3k
views
Gröbner bases in TCS?
Does anyone know of interesting applications of Gröbner bases to theoretical computer science?
Gröbner bases are used to solve multi-variate polynomial equations, an NP-hard problem in general. I was ...
- 11k
28
votes
6
answers
3k
views
Alternative proofs of Schwartz–Zippel lemma
I'm only aware of two proofs of Schwartz–Zippel lemma. The first (more common) proof is described in the wikipedia entry. The second proof was discovered by Dana Moshkovitz.
Are there any other ...
- 3,674