Questions tagged [algebraic-complexity]

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1answer
174 views

Size of Formulas with no negative sign for Matrix Permanent

What is the best lower bound for algebraic formulas for Permanent of a matrix given that the formulas have no negative sign? Is there an exponential lower bound known for such formulas and what would ...
5
votes
1answer
170 views

Complexity of counting integer roots of multivariate polynomials in a polyhedron?

Deciding integer roots of multivariate polyomials is undecidable. However what is known about counting integer roots of multivariate polynomials in $\mathbb Z[x_1,\dots,x_m]$ with both $m$ and total ...
5
votes
1answer
566 views

Permanent as projection of determinant and another permanent

I am looking for explicit examples where permanent of a given matrix $A$ is given by a determinant of a larger matrix $B$ (projection in the sense of Valiant). Is there any reference where I can find ...
5
votes
1answer
225 views

Questions about P vs NP and geometric complexity theory

Reading through various papers on geometric complexity theory (GCT), there is one thing, which pops up, while claimed in various places, that it is an approach to P vs NP, all the results seems to ...
5
votes
1answer
432 views

Classical Matrix-Vector multiplication Complexity of standard matrices

Why are standard unitary transforms such as the Fourier and the Hadamard transforms believed to have a multiplicative complexity (number of multiplications) of $O(n^{1+\delta_{m}})$ and an additive ...
5
votes
1answer
607 views

Lower bounds for Polynomials computing the boolean functions

Expressing a boolean function $f$ $:\{ 0,1 \}^{n} \rightarrow \{0,1 \}$ using a polynomial $P(x_{1},...,x_{n})$, where $x_{1},...,x_{n}$ may be integer, finite fields, or other fields. One of the most ...
5
votes
2answers
377 views

Complexity of algorithm to test if a graph is asymmetric

Counting the order of automorphism group of a graph is polynomial-time equivalent to graph isomorphism problem. But if we just want to know if the order is greater than 1, what is the complexity of ...
5
votes
0answers
233 views

What is the status of Determinantal Complexity of Permanent

Recently Landsberg and Ressayre announced an exponential lower bound for permanent's determinantal complexity assuming certain symmetry conditions. What is the status of the problem of Permanent's ...
4
votes
1answer
180 views

Is $GCT$ necessarily a negative result program?

$GCT$ is a candidate program to separate permanent and determinant through symmetries. If indeed permanent and determinant can be handled in similar complexity class would $GCT$ be a program which can ...
4
votes
2answers
1k views

Complexity of the inverse modulo a composite number

Supposing $M$ is a composite number and supposing $a$ is an integer such that $a^{-1}\mod M$ exists, can we compute $a^{-1} \bmod M$ by using $O(\log^{b}(M))$ ring operations in the RAM model, where $...
3
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1answer
239 views

Convolution without FFT

What is the best upper and lower bound known for convolution without FFT? Is FFT proven to be essential for time complexity reduction? Is cancellation essential as well?
3
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0answers
46 views

Equal degree factoring of homogeneous polynomials over $\Bbb Q[x_1,\dots,x_n]$?

Given $f(x_1,\dots,x_n)\in\Bbb Q[x_1,\dots,x_n]$ of form $\prod_{i=1}^df_i(x_1,\dots,x_n)$ where each of $f,f_i$ are homogeneous and each $f_i$ is irreducible what is the best technique to factor such ...
2
votes
2answers
109 views

IPS upper bound for subset sum axiom

I am reading the following paper Michael A. Forbes, Amir Shpilka, Iddo Tzameret, Avi Wigderson ,"Proof Complexity Lower Bounds from Algebraic Circuit Complexity", 2016. IPS is defined as follows: ...
2
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0answers
128 views

Matrix multiplication when one matrix is fixed

Let $A$ be a fixed positive entried integer matrix of size $a\times n$ with $\ell$ bits per entry One is allowed to pre-process this matrix as appropriate. Given another positive integer entried $B$...
2
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0answers
110 views

On $\Sigma \Pi \Sigma \Pi(2,r)$-circuits

As I understand from the survey "Progress on Polynomial Identity Testing - II" a polynomial-time algorithm for solving PIT for $\Sigma \Pi \Sigma \Pi (2, r)$ is unknown. However, there exists paper ...
2
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0answers
122 views

Complexity of a particular determinant

Suppose we have an $n\times n$ matrix $A$ with non-negative integer entries such that $\mathsf{Tr}(A^i)=0$ at every $i\in\{1,2,\dots,n-2,n-1\}$ and $\mathsf{Tr}(A^n)\neq0$, then from Trace-Determinant ...
1
vote
1answer
149 views

Morgenstern's Theorem

Morgenstern proves a $\Omega(n\log n)$ lower bound for Fourier transform in the bounded coefficient model. Let $x=[x_1,x_2,\ldots,x_n]'$ be given vector and $F$ be Fourier transform matrix. It is ...
1
vote
1answer
112 views

Hitting set of very restricted linear forms

We say that $f\in\mathbb{Z}[x_{1},\dots,x_{n}]$ is a {-1,0,1}-linear form if $f=\sum_{i\in S}x_{i}-\sum_{i\in T}x_{i}$ where $S,T\subseteq[n]$. A hitting set $H\subseteq\mathbb{Z}^{n}$ for {-1,0,1}-...
1
vote
1answer
158 views

What is the computational complexity of solutions over $\mathbb{Q}$ of polynomial equation with coeffiecents over $\mathbb{Z}$

What is the complexity of the following problem? (e.g. best-known running time, space, best upper bound in terms of complexity classes, etc.) Input: A multivariate polynomial $f$ with coefficients ...
1
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0answers
926 views

How can I understand the Coppersmith–Winograd algorithm?

I want to do research on matrix multiplication algorithms. I glanced at the Coppersmith-Winograd algorithm paper, but I didn't understand anything. How can I complete the background to read this paper?...
1
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0answers
101 views

Complexity of planted root of a system of quadratic homogeneous polynomials?

Given homogeneous degree $2$ randomly chosen polynomials $f_1,\dots,f_{m}$ in $\mathbb Z[x_1,\dots,x_n,y_1,\dots,y_n]$ each with only monomials $x_iy_j$ with condition that the system $f_1=\dots=f_{m}=...
1
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0answers
21 views

Worst case polynomial in elimination theory under rank conditions?

Given $n$ polynomials $h_1(x_1,\dots,x_{2n}),\dots,h_{2n}(x_1,\dots,x_{2n})\in\mathbb Z[x_1,\dots,x_{2n}]$ where each of $h_1(x_1,\dots,x_{2n}),\dots,h_{2n}(x_1,\dots,x_{2n})$ is homogeneous of degree ...
1
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0answers
68 views

Relating classes of computational complexity to finding solution to classes of algebraic equations [closed]

Having related classes of computational complexity to finding solution to classes of algebraic equations, we may relate classes of computational complexity to algebraic geometry or complex geometry,...
1
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0answers
89 views

What is the relation between computational complexity of algebraic number and computational complexity to find the solution to algebraic equation?

Suppose $\alpha$ is algebraic number, and we have the algorithm with lowest computational complexity to output it, and $f(x)=0$ is algebraic polynomial with $\alpha$ as a root. If an algorithm which ...

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