Questions tagged [polynomials]

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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 ...
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849 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 ...
368 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 ...
428 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 ...
297 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 ...
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Evaluating symmetric polynomials

Let $f:\mathbb{K}^n \to \mathbb{K}$ be a symmetric polynomial, i.e., a polynomial such that $f(x)=f(\sigma(x))$ for all $x \in \mathbb{K}^n$ and all permutations $\sigma \in S_n$. For convenience, we ...
344 views

Is there a P-complete problem on diophantine equations?

In general deciding whether a diophantine equation has any integer solutions is equivalent to the halting problem. I believe that deciding if a quadratic diophantine equation has any solution is NP-...
409 views

How "hard" is it to maximize a polynomial function subject to linear constraints?

General Problem Suppose we have a multivariate polynomial function $f(\mathbf{x})$, and several linear functions $\ell_i(\mathbf{x})$. What is known about the complexity of solving the following ...
170 views

Maintaining the value of a polynomial over a dynamically updated input

Let $P(x_1, x_2, \ldots, x_n)$ be a polynomial over a fixed finite field. Suppose we are given the value of $P$ on some vector $y \in \{0,1\}^n$ and the vector $y$. We now want to compute the value ...
371 views

On derandomizing polynomial identity testing

In polynomial identity testing we seek a deterministic algorithm to infer equality of two polynomials $g,h\in\Bbb Z[x_1,\dots,x_n]$. Derandomizing known efficient randomized algorithms and producing ...
526 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 ...
293 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 ...
240 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 ...
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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 ...
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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 ...
299 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 ...
787 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 ...
272 views

What do we know about checking real-stability of multivariate complex polynomials?

Given a polynomial $p : \mathbb{C}^n \rightarrow \mathbb{C}$ it is to be called "real-stable" if (1) all its coefficients are real and (2) if it has no roots such that all the coordinates of the root ...
270 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 ...
356 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 ...
191 views

Fast algorithm for composing Boolean polynomials

Question: Is there a practically fast algorithm to compose Boolean polynomials ($\mathbb{F}_2[x]$) modulo a fixed Boolean polynomial? Background: I would like random access within Vigna's xorshift128+...
192 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 $w_{ij}$ ...
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Complexity of extracting a coefficient of a polynomial in multiple variables

I'm looking for efficient algorithms for problems of the following type: Let's say we have the variables $x_1,...,x_n$. Over these variables, we are given a function $p_1\cdot ... \cdot p_m$, ...
111 views

Algebraic dependence of roots of irreducibles over a finite field

I asked this question in Math SE too, but I have since modified it to make it more suited here. Also, in hindsight, the question itself was more algorithmic and was a better fit here. https://math....