# Questions tagged [algebraic-complexity]

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223 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 ...
177 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 ...
895 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?...
127 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 $B$...
143 views

### What are some examples of algorithmic applications of noncommutative rational identity testing?

The problem of polynomial identity testing (PIT) is known to be in $\mathsf{RP}$, but not known to be in $\mathsf{P}$. The related problem of noncommutative rational identity testing (NCIT) is known ...
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### 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 ...
169 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 ...
56 views

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### 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,...
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: ...
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 ...
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 ...
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 ...
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 ...
292 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 ...
517 views

### Riemann Hypothesis and Complexity Theory

It is known that "Assuming the generalized Riemann hypothesis (GRH) if VP = VNP then PH collapses to second level". Why would one think of a relation between VP,VNP and the Riemann hypothesis. Where ...
686 views

### VC dimension of polynomials over tropical semirings?

As in this question, I am interested the $\mathbf{BPP}$ vs. $\mathbf{P}$/$\mathrm{poly}$ problem for tropical $(\max,+)$ and $(\min,+)$ circuits. This question reduces to showing upper bounds for the ...
263 views

### Checking whether two quadratic equations have a common zero

Given two quadratic equations (with integer coefficients): $x^T A_1 x+ b^T_1 x + c_1=0$ and $x^T A_2 x+ b^T_2 x + c_2=0$ The problem is to decide whether they have a common zero. Here $x$ is a ...
151 views

### Recognition of a primitive root

Adleman and McCurley published a paper in 1994 called "Open problems in number theoretic complexity, II" (http://ww.cstheory.com/papers/open.ps.gz) Problem 18 of this list of open problems is about ...
181 views

### Implications of a recent negative result to geometric complexity

A paper was posted in arxiv http://arxiv.org/pdf/1512.03798.pdf titled 'Rectangular Kronecker coefficients and plethysms in geometric complexity theory' by Christian Ikenmeyer and Greta Panova with ...
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 ...
224 views

### Sorting using ring operations

Sorting is in $\mathsf{NP}$. Given a sorted list, it is trivial to check sortedness in linear time. Is there any evidence sorting of elements from an ordered gcd domain(eg: $\Bbb Z$) cannot be done ...
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 ...
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### 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 ...
2k views

### A course for learning algebraic complexity

I want to learn about algebraic algorithms and complexity thoery. In particular, I am interested in PIT. Is there a set of lecture notes, books, papers and surveys for students who have read standard ...
217 views

### Checking if a polynomial factors into linear factors

Let $f\in\mathbb{Q}[x_{1},x_{2},\ldots,x_{n}]$ be a polynomial given by an arithmetic circuit $C$ of size $s$. Given $C$ as the input, is there a deterministic algorithm to check whether all the ...
157 views

### 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 ...
748 views

### Expressing Determinant as Permanent

One major problem in TCS is the problem of expressing a permanent as a determinant. I was reading Agrawal's paper Determinant Versus Permanent and in one paragraph he claims the reverse problem is ...
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 ...
378 views

### Uniformity vs. nonuniformity in algebraic complexity theory

I understand that the study of Boolean circuits and nonuniform complexity classes was introduced to (hopefully) prove separation of uniform complexity classes. In this sense, nonuniform computational ...
571 views

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### 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 ...
288 views

### Commutative matrix multiplication algorithms

What is known about commutative algorithms like Winograd algorithm and its variants for Matrix Multiplication? Why is there not much study on them? Can they be asymptotically as efficient as Non-...
1k views

### Determinants and Matrix Multiplication - Similarity and differences in algorithmic complexity and arithmetic circuit size

I am trying to understand the relation between algorithmic complexity and circuit complexity of Determinants and Matrix Multiplication. It is known that the determinant of an $n\times n$ matrix can ...
721 views

### Cancellation and determinant

Berkowitz algorithm provides a polynomial size circuit with logarithmic depth for determinant of a square matrix using matrix powers. The algorithm implicitly uses cancellation. Is cancellation ...