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8
votes
1answer
110 views

Randomized identity-testing for high degree polynomials?

Let $f$ be an $n$-variate polynomial given as an arithmetic circuit of size poly$(n)$, and let $p = 2^{\Omega(n)}$ be a prime. Can you test if $f$ is identically zero over $\mathbb{Z}_p$, with time ...
2
votes
0answers
42 views

Efficient Shamir secret sharing reconstruction

Shamir's secret sharing scheme is a well known way to convert a secret into a polynomial and distribute points in this polynomial. Some of these points can then be regrouped to reconstruct the ...
2
votes
2answers
129 views

An upper bound over the number of bipolar orientations for a regular graph

Given a $k$-regular graph $G$, the number of acyclic orientations $Acy(G)$ is $\chi(-1)$ where $\chi$ is the chromatic polynomial of $G$. How many bipolar orientations does $G$ have? Is there an ...
3
votes
2answers
159 views

Root finding in [0,1]

I am interested in the problem of finding a real root of a polynomial equation $f(x)=0$ where $f(x)=\sum_{i=0}^n a_ix^i$. Is it possible to give a reduction, i.e, to compute a different polynomial $g$ ...
4
votes
0answers
74 views

The rank-polynomial of a graded poset

Let $P$ be a graded poset with rank function $r$. We may then define its rank-polynomial as: $R_P(q) = \sum_{x \in P} q^{r(x)}$. This definition can be applied to several interesting posets, for ...
5
votes
1answer
71 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 ...
21
votes
2answers
1k views

Representing OR with polynomials

I know that trivially the OR function on $n$ variables $x_1,\ldots, x_n$ can be represented exactly by the polynomial $p(x_1,\ldots,x_n)$ as such: $p(x_1,\ldots,x_n) = 1-\prod_{i = ...
0
votes
1answer
82 views

What are some efficient algorithms for determining if a system of quadratic multivariate polynomials have a solution?

I know that in the general case it isn't efficient. However, I'm wondering if there are any good techniques in the quadratic case over the reals, specifically if there is a polynomial time solution.
2
votes
0answers
115 views

What are some efficient algorithms for finding the solutions to a quadratic multivariate polynomial?

Based on this question, there's an efficient algorithm to determine whether a quadratic multivariate polynomial has a root. What are some algorithms to enumerate these solutions? I'm interested in ...
6
votes
1answer
120 views

Complexity of convolution in the max/plus ring

We can do convolution in $O(nlgn)$ for plus/multiply polynomials with fft. However the approach doesn't seem very generalisable to rings in general. Has there been any progress over the naive ...
6
votes
4answers
430 views

What are some efficient algorithms for determining if a quadratic multivariate polynomial has a solution?

I know that in general, multivariate polynomial satisfiability is equivalent to 3-SAT; however, I'm wondering if there are any good techniques in the quadratic case, specifically if there is a ...
2
votes
0answers
92 views

Geometry on a space of polynomial functions

I am considering some geometric concepts in a space of functions.But I am not sure the concept I consider is already defined in some references. Let $P_{1},...,P_{N}:\mathbb{F}_{2}^{n}\rightarrow ...
2
votes
0answers
64 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 ...
14
votes
0answers
203 views

Exponential-time factorization of polynomials

Let an explicit field be a field for which equality is decidable (in some standard model of computation). I am interested in the factorization of univariate polynomials over an explicit field. It is ...
1
vote
0answers
67 views

Verifying the minimum degree of a geometric predicate

Let $F = (f_1, \ldots, f_n)$ be a sequence of multivariate polynomials $f_i : \mathbb{R}^d \to \mathbb{R}$, and $g : \{0,1\}^n \to \{0,1\}$ a boolean function. Say the composition ...
8
votes
0answers
174 views

Counting small terms in a determinant calculation over polynomials (counting spanning trees by weight)

I have a $n\times n$ matrix $A$. It's terms are $a_{ij}=-x^{w_{ij}}$ if $i\neq j$ and $a_{ii}=\sum_{j=0}^{n+1} x^{w_{ij}}$ on the diagonal. The matrix is symmetric as $w_{ij}=w_{ji}$. Numbers $w_{ij}$ ...
9
votes
3answers
397 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 ...
20
votes
0answers
427 views

Approximate degree of $\textrm{AC}^0$

EDIT (v2): Added a section at the end on what I know about the problem. EDIT (v3): Added discussion on threshold degree at the end. Question This question is mainly a reference request. I don't ...
7
votes
1answer
224 views

What are some results on algorithms that estimate polynomials over a given set of points?

There seem to be many randomized algorithms for polynomial identity testing, checking whether or not a given polynomial is zero. Are there any results of algorithms that do some sort of estimation of ...
13
votes
2answers
282 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 ...
15
votes
0answers
320 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 ...
2
votes
1answer
129 views

Parallel algorithms to find the optimum of polynomials

Are there any non-trivial parallel algorithms to find the optimum (local or global) of polynomials? By trivial, I mean something which is an obvious application of a serial algorithm. For example, one ...
7
votes
0answers
123 views

Computing the Fourier-Walsh coefficients of an arithmetic circuit

Given an $f$-fan in arithmetic circuit of depth $d$ over $GF(q)$ (for some prime $q$) and the set of variables $(x_1,\ldots,x_n)$, what is the state of the art upper bound for calculating the ...
6
votes
1answer
197 views

Results in computer science via the Stepanov method

I recently attended a Workshop on Pseudorandomness in Chennai Mathematical Institute on Pseudorandomness. Venkat Guruswami made the following beautiful statement in passing during his talk (on coding ...
8
votes
1answer
386 views

Systematic studies of sum of quadratic polynomials squared

I'm wondering if there exists systematic studies of sums of quadratic forms squared, similar to the quadratic forms, which is practically reflected in eigenvalue decomposition (that has huge practical ...
34
votes
10answers
1k 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 ...
16
votes
2answers
579 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 ...
5
votes
0answers
189 views

Is there an interpretation of the degree of the polynomial computed by the arithmetization of a boolean formula?

If I understand this correctly, arithmetizating a formula is a way of "interpolating" a polynomial in such a way that polynomial evaluated on 0's and 1's corresponds to "unsatisfiable" if it is a ...
11
votes
1answer
230 views

Explicit polynomials in 1 variable with superlogarithmic circuit complexity lower bounds?

By counting arguments, one can show that there exist polynomials of degree n in 1 variable (i.e., something of the form $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0)$ which have circuit complexity n. ...
3
votes
1answer
175 views

Applications of association schemes to complexity theory and other TCS

An association scheme is defined as a pair $(V, R_0,R_1, \ldots,R_{n+1})$ of a set $V$ and relations $R_i$ on $V$ such that $(x,y) \in R_i$ implies $(y,x) \in R_i$ for all $x, y \in V$. $R_0 = \{ ...
12
votes
2answers
294 views

Evaluating the multilinearization of an arithmetic circuit?

Let $p(x_1,\ldots,x_n)$ be a multi-variate polynomial with coefficients over a field $F$. The multilinearization of $p$, denoted by $\hat{p}$, is the result of repeatedly replacing each $x_i^d$ with ...
27
votes
2answers
591 views

Polynomial method for complexity results

Polynomial methods, say Combinatorial Nullstellensatz and Chevalley–Warning theorem are powerful tools in additive combinatorics. By representing a problem with proper polynomials, they can guarantee ...
0
votes
1answer
508 views

How to compute ROOK Polynomials for NxM Matrices [closed]

How to compute ROOK Polynomials for NxM Matrices for k objects ?
5
votes
1answer
454 views

What is the running time of taking a limit?

I'm interested in finding the running time(s) for determining mathematical limits. For instance, $\lim_{x \to 2} \frac{1}{x} = \frac{1}{2}$. I'd like to know more about algorithms for determining ...
2
votes
0answers
250 views

Some problems involving polynomials of public and private variables over GF(2).

Suppose there are a set of low degree (less than some degree $z$) polynomials $P_0, P_1, ..., P_k$ each of which is defined over two types of variables, red variables ${v_r}_0, {v_r}_1, ..., {v_r}_n$ ...
34
votes
1answer
2k views

Multiplying n polynomials of degree 1

The problem is to compute the polynomial $(a_1 x + b_1) \times \cdots \times (a_n x + b_n)$. Assume that all coefficients fit in a machine word, i.e. can be manipulated in unit time. You can do $O(n ...