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# Questions tagged [algebra]

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### 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 ...
518 views

102 views

### Equivalence relations on strongly regular graphs with same parameters

Let $\mathcal{S}$ be the set of all strongly regular graphs with parameter $(n, k, \lambda, \mu)$. Are there any (interesting) equivalence relations defined on this set? My motivation is to approach ...
204 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 ...
120 views

### count number of i such that ( (a*i+b) mod p) mod k == l

How to determine the number of $i$'s as fast as possible such that $$1\le i \le L and ((ai+b)\mod p) \mod k = l$$ where $p$ is a prime number, $1\lt a, b\lt p-1$, and $l \lt k \lt L \lt p$. This ...
1k views

### How to go about proving the basic operators in relational algebra are independent of each other?

The five basic operator select, project, cross, union and diff in relational algebra are independent of each other. I'm trying to formally prove this statement but can only progress for cross product ...
849 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 ...
366 views

### What's the bias of random polynomials with low degree over GF(2)?

I have a question concerning low-degree polynomials and probability: What is the (assyptotic behavior of the) probability that a random* polynomial, $p$, over GF(2), with degree $\le d$ and n ...
293 views

### Sufficient conditions to guarantee unique fixpoint (not unique least/greatest fixpoint) for monotone functions on complete lattice

Tarski's fixpoint theorem states, that the fixpoints of a monotone operator on a complete lattice is a complete lattice. By consequence, we have a unique greatest fixpoint and unique least fixpoint ...
19k views

### Uses of algebraic structures in theoretical computer science

I'm a software practitioner and I'm writing a survey on algebraic structures for personal research and am trying to produce examples of how these structures are used in theoretical computer science (...
359 views

### Looking for an operator on polynomials

I have a small, self-contained, math question, whose motivation is from theoretical computer science (specifically, list decoding of algebraic codes, derivative/multiplicity codes, etc). I wonder ...
2k views

### Data Structure isomorphisms

Disclaimer: I am not a CS theorist. Coming from abstract algebra, I'm used to dealing with things that are equal up to a isomorphism - but I'm having a trouble translating this concept to data ...
287 views

### Smallest representatives of a quotient by an equivalence relation

Background Let $\mathcal{A}=(Q,\Sigma,\delta,q_0,F)$ be a minimal DFA for a regular language $L$ such that $|Q|=n$, and let $\equiv_L$ be the relation given by x\equiv_Ly\text{ iff for all $u$: }xu\...
336 views

2k 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 ...
312 views

### Are there any 'graphical' algebras that can describe the 'shape' of graphs?

One of the main problems in graph enumeration is determining the 'shape' of a graph, e.g. the isomorphism class of any particular graph. I am fully aware that every graph can be represented as a ...
361 views

### Is there any research on Turing machines with transition relation homomorphic to given algebraic structure?

A Turing machine is defined as a structure $TM(L,Q,T)$, where $L,Q$ are sets of symbols and internal states of TM respectively, and T is a transition relation: $T: L \times Q \to L \times Q$ for ...
778 views

### Computing sum of sparse polynomials squared in O(n log n) time?

Suppose we have polynomials $p_1,...,p_m$ of degree at most $n$, $n>m$, such that the total number of nonzero coefficients is $n$ (i.e., the polynomials are sparse). I am interested in an efficient ...
3k views

### The quad-edge data structure (Delaunay/Voronoi)

2 questions for the computational geometers or algebraists: I am just beginning to dive into computational geometry and I am loving it =) I am attempting to read the famous article by Guibas and ...
421 views

### Field extensions in CS

A field is a set with two binary operations called addition and multiplication satisfying various axioms. Wikipedia article: Field_(mathematics) A field extension is when you add a new element and ...
599 views

### Formal representation of rings in computations

While reading a paper about using algebraic methods to detect some induced subgraphs, it appears that edge ideal is an important tool connecting commutative algebra and graph theory. Since I'm not ...
674 views

### Is Witten's new method of quantization useful for geometric complexity theory? [closed]

The Kempf-Ness theorem (see e.g. arXiv:0912.1132) - that the algebraic quotient of geometric invariant theory is also a symplectic quotient - suggests (to me) that certain physical constructions used ...
2k 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 ...
282 views

### What is/are the lower bounds for finding a something akin to complex residue?

Given a function $\sum_{i=-N}^N{c_i x^i}$: $f(x) \equiv \sum_{i=-N}^N{c_i x^i}$ where $c_i$ is an integer; $0 \le c_i \le a$ for some $a$. The constant $c_0$ is desired, and we start with only \$f(x)...
565 views

### Hardness Guarantees for AES

Many public-key cryptosystems have some kind of provable security. For example, the Rabin cryptosystem is provably as hard as factoring. I wonder whether such kind of provable security exists for ...