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44
votes
4answers
2k views

What algorithm in algebraic geometry should I work on implementing?

This summer my wife and one of my friends (who are both programmers and undergraduate math majors, but have not learned any algebraic geometry) want to learn some algebraic geometry from me, and I ...
35
votes
1answer
1k views

improving known bounds for Pierce expansions; cash prize

Here's a problem that I thought of back in 1978 or so, and only a little progress has been made on it since then. I still think about it from time to time, but probably not that many people have ...
33
votes
4answers
7k views

How hard is it to compute the number of prime factors of a given integer?

I asked a related question on this mathoverflow thread. That question was promptly answered. This is a natural followup question to that one, which I decided to repost since that question is answered. ...
31
votes
7answers
2k views

What is the time complexity of computing sin(x) to t bits of precision?

Short version of the question: Presumably, it's poly$(t)$. But what polynomial, and could you provide a reference? Long version of the question: I'm sort of surprised to be asking this, because ...
30
votes
2answers
2k views

Does anyone want a pretty Maass form?

A few months ago, I was curious about some properties of Maass cusp forms, of nonabelian arithmetic origin. As a result, I went through a somewhat predictable process of finding a totally real $A_4$ ...
28
votes
3answers
2k views

Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable?

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$, decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective, respectively, injective? -- And ...
28
votes
1answer
3k views

An edge partitioning problem on cubic graphs

Hello everyone, I already asked this question on the TCS Stack Exchange, but it has not been resolved yet. Maybe readers of this forum will have other ideas or information, although I suspect that ...
27
votes
3answers
2k views

Can assignment solve stable marriage?

This is an excellent question asked by one of my students. I imagine the answer is "no", but it doesn't strike me as easy. Recall the set up of the stable marriage problem. We have $n$ men and $n$ ...
26
votes
4answers
2k views

Why is “P vs. NP” necessarily relevant?

I want to start out by giving two examples: 1) Graham's problem is to decide whether a given edge-coloring (with two colors) of the complete graph on vertices $\lbrace-1,+1\rbrace^n$ contains a ...
26
votes
1answer
413 views

Decidability of equality of expressions built using 1,+,-,*,/,^

Consider expressions built using number $1$, arithmetical operators $+, -, *, /$ and exponentiation ^ (in case of multiple values, the principal value is assumed, the same way as it implemented in ...
25
votes
3answers
1k views

Is it decidable whether or not a collection of integer matrices generates a free group?

Suppose we have integer matrices $A_1,\ldots,A_n\in\operatorname{GL}(n,\mathbb Z)$. Define $\varphi:F_n\to\operatorname{GL}(n,\mathbb Z)$ by $x_i\mapsto A_i$. Is there an algorithm to decide whether ...
25
votes
2answers
919 views

Groebner basis with group action

At one point my advisor, Mark Haiman, mentioned that it would be nice if there was a way to compute Groebner bases that takes into account a group action. Does anyone know of any work done along ...
23
votes
19answers
3k views

What is the easiest randomized algorithm to motivate to the layperson?

When trying to explain complexity theory to laypeople, I often mention randomized algorithms but seemingly lack good examples to motivate their usage. I often want to mention primality testing but ...
23
votes
4answers
2k views

Can Gröbner bases be used to compute solutions to large, real-world problems?

In particular, suppose I have an affine algebraic variety over $\mathbb{R}^n$ described by generators of a radical ideal $I$ and I want to find (perhaps not all of the) points in the variety. Several ...
21
votes
8answers
9k views

Fast Matrix Multiplication

Suppose we have two $n$ by $n$ matrices over particular ring. We want to multiply them as fast as possible. According to wikipedia there is an algorithm of Coppersmith and Winograd that can do it in ...
20
votes
3answers
1k views

Can one measure the infeasibility of four color proofs?

Terms like "impractical" and "unfeasible" are used to say the Robertson, Sanders, Seymour, and Thomas proof of the four color theorem needs computer assistance. Obviously no precise measure is ...
20
votes
4answers
1k views

Securing privacy of “who communicates with whom” under Orwell-like conditions

Assume that there is a big and powerful country with an information-greedy secret service which has backdoors to all internet nodes throughout the world which permit him to observe all exchanged data ...
19
votes
0answers
343 views

Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them. Define the set $\mathcal{E}$ of ...
18
votes
1answer
2k views

How fast can we *really* multiply matrices?

Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.8})$ for the multiplication of two $n \times n$ matrices. However, the constant is so large that ...
18
votes
3answers
2k views

Satisfiability of general Boolean formulas with at most two occurrences per variable

(If you know basics in theoretical computer science, you may skip immediately to the dark box below. I thought I would try to explain my question very carefully, to maximize the number of people that ...
17
votes
0answers
153 views

A small unavoidable collection of subgraphs

What is the smallest number S(k,n) of unlabeled graphs on k vertices such that every simple graph on n vertices contains at least one of these as an induced subgraph? I'd like to avoid exhaustive ...
16
votes
2answers
3k views

Generalization of the shakehands/condom puzzle?

The classic handshake puzzle goes something like this: "Given that everyone has a different skin disease, how can you safely shake hands with 3 people when you have only 2 gloves?" Its common ...
16
votes
5answers
3k views

Fastest Algorithm to Compute the Sum of Primes?

Can anyone help me with references to the current fastest algorithms for counting the exact sum of primes less than some number n? I'm specifically curious about the best case running times, of ...
16
votes
9answers
2k views

Determine if circle is covered by some set of other circles

Suppose you have a set of circles $\mathcal{C} = \{ C_1, \ldots, C_n \}$ each with a fixed radius $r$ but varying centre coordinates. Next, you are given a new circle $C_{n+1}$ with the same radius ...
16
votes
6answers
3k views

Algorithms for finding rational points on an elliptic curve?

I am looking for algorithms on how to find rational points on an elliptic curve $$y^2 = x^3 + a x + b$$ where $a$ and $b$ are integers. Any sort of ideas on how to proceed are welcome. For example, ...
16
votes
5answers
2k views

duplicate detection problem

Restated from stackoverflow: Given: array a[1:N] consisting of integer elements in the range 1:N Is there a way to detect whether the array is a permutation (no duplicates) or whether it has any ...
16
votes
3answers
842 views

Easy functions ?

Let $f$ be an analytic function, and suppose that we want to compute $f(x)$. The input consists of the digits of $x$ and the output of a rational number approximating $f(x)$. A function $f$ is called ...
16
votes
2answers
2k views

Inverting the totient function

For what values of $n$ does the equation $\phi(x) = n$ have at least one solution? Is there any efficient way to check it for a given $n$? It obviously has no solutions for odd $n$. And the smallest ...
15
votes
12answers
2k views

Lower Bounds in Theoretical Computer Science

Besides the classical: you can't do comparison sort with faster than (n logn); what are other lower bounds we know of for algorithms? I can't seem to dig them up via google scholar, yet they must ...
15
votes
9answers
7k views

Algorithm for finding the volume of a convex polytope

It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ...
15
votes
4answers
773 views

Proving univariate polynomials (defined by sums, binomial coeffs, etc.) are nonnegative: is it 'routine'?

My colleagues and I are working on a project related to an old paper of C. Borell and we have boiled it down to the following problem: Show, for all integers $1 \leq i \leq k$, that the univariate ...
15
votes
4answers
2k views

FFTs over finite fields?

I'm trying to understand how to compute a fast Fourier transform over a finite field. This question arose in the analysis of some BCH codes. Consider the finite field $F$ with $2^n$ elements. It is ...
14
votes
10answers
6k views

What are the Applications of Hypergraphs

Hypergraphs are like simple graphs, except that instead of having edges that only connect 2 vertices, their edges are sets of any number of vertices. This happens to mean that all graphs are just a ...
14
votes
9answers
2k views

How can I generate random permutations of [n] with k cycles, where k is much larger than log n?

I've been thinking a lot lately about random permutations. It's well-known that the mean and variance of the number of cycles of a permutation chosen uniformly at random from Sn are both ...
14
votes
5answers
998 views

Algorithm generalizing continued fractions for non-quadratic algebraic numbers

The continued fraction algorithm generates an integer sequence which terminates for a rational number, is periodic for the roots of irreducible integer quadratics, and is non-periodic for other ...
14
votes
3answers
1k views

does the “convolution theorem” apply to weaker algebraic structures?

The Convolution Theorem is often exploited to compute the convolution of two sequences efficiently: take the (discrete) Fourier transform of each sequence, multiply them, and then perform the inverse ...
13
votes
3answers
5k views

Finding a cycle of fixed length

Is there any result about the time complexity of finding a cycle of fixed length k in a general graph? All I know is that Noga Alon et al. use the techinique called "color-coding", which has a running ...
13
votes
3answers
1k views

When are (finite) simplicial complexes (smooth) manifolds?

Hi, is there an algorithm that determines if a given simplicial complex is a.) a manifold b.) a smooth manifold c.) homotopy equivalent to a manifold d.) a real algebraic variety ?
13
votes
2answers
646 views

Are the axioms for higher category-theory effectively computable?

I ask this, although I don't conduct any research in the area, or even plan to. -- There seems to be general agreement that the axioms for higher categories grow very rapidly in complexity as the ...
13
votes
1answer
2k views

Distinct numbers in multiplication table

Consider multiplication table for numbers $1,2,\cdots, n$. How many different numbers are there? That is how many different numbers of the form $ij$ with $1 \le i, j \le n$ are there? I'm interested ...
13
votes
6answers
4k views

Good algorithm for finding the diameter of a (sparse) graph?

My question on Stack Overflow was recently tagged "math". Despite a bounty, it never received a satisfactory answer, so I thought I would ask it here: I have a large, connected, sparse graph in ...
13
votes
3answers
2k views

Computing the maximum salary

To motivate my question, I will describe a related problem and then give a solution to it. My question will then be a variant of this problem. N individuals sit around a table and want to compute the ...
13
votes
2answers
880 views

What is the most compelling reason to believe Church's thesis? [closed]

Church's thesis states that the Turing machine is a universal model of computation. What is the most compelling argument supporting this assertion?
13
votes
2answers
899 views

Subfields of a function field

Is there an algorithm for generating (some or all) subfields of a certain genus of a given function field (even a random one,I mean for example generating a random elliptic subfield of a certain given ...
13
votes
2answers
916 views

Seeking proof for linear algebra constraint problem.

Given a symmetric real matrix with a zero diagonal $M$, I am trying to find a diagonal matrix $D$, such that the matrix $M + D$ is positive definite, and $(M+D)^{-1}$ has a diagonal consisting of all ...
13
votes
2answers
1k views

Detecting almost-primes quickly

There are many fast algorithms (deterministic and probabilistic) for detecting primality. Are there any fast algorithms (probabilistic ones allowed) known for detecting whether a number is the product ...
13
votes
0answers
2k views

Minimum tiling of a rectangle by squares

Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes). Is there an efficient way to calculate this?
12
votes
3answers
484 views

Recognizing the 4-sphere and the Adjan--Rabin theorem

The problem of recognizing the standard $S^n$ is the following: Given some simplicial complex $M$ with rational vertices representing a closed manifold, can one decide (in finite time) if $M$ is ...
12
votes
1answer
2k views

Evidence for integer factorization is in $P$

Peter Sarnak believes that integer factorization is in $P$. It is a well-known open problem in TCS to identify the real complexity class of integer factorization. Take a look at this link for Peter ...
12
votes
4answers
2k views

How Does One Find the “Loneliest Person on the Planet”?

I'm looking for the algorithm that efficiently locates the "Loneliest Person on the Planet", where "loneliest" is defined as: Maximum minimum distance to another person -- that is, the person for ...