Questions tagged [polynomials]

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3
votes
0answers
173 views

Testing emptiness property complexity in Sum of Squares Proof systems

Take the set $$\mathcal T=\{f_1(x_1,\dots,x_n)=\dots=f_m(x_1,\dots,x_n)=0, h_1(x_1,\dots,x_n)\geq a_1,\dots,h_t(x_1,\dots,x_n)\geq a_t\}$$ where $$h_1(x_1,\dots,x_n),\dots,h_t(x_1,\dots,x_n)\in\mathbb ...
1
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0answers
166 views

Can a hash preimage be used to amplify BPP probabilities?

Suppose we are given a (univariate) polynomial $P$ of degree $d$, and we wish to determine if $P$ is identically $0$. A standard way to do this is to use a classical PRG to randomly sample a number $...
1
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0answers
80 views

Switching lemma for polynomials over $\mathbb{F}_2$

Suppose $f$ is in $\mathbb{F}_2[x_1,...,x_n]$ with total degree $d$. Q. Is there any kind of switching lemma or restriction lemma in which by applying the lemma on $f$ we can reduce the total ...
6
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2answers
224 views

Razborov-Smolensky polynomial argument on $\textrm{ACC}[q]$ where $q$ is a prime power

It seems to be a folklore that we can "handle" $\textrm{ACC}[q]$ circuits not only for prime $q$ but prime power $q$. For example, authors of this paper say that ... any constant depth circuit ...
18
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3answers
776 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 ...
10
votes
1answer
500 views

Complexity of convolution in the max/plus ring

We can do convolution in $O(n\log n)$ 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 $O(...
0
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0answers
54 views

Which research fields deal with this variant definition of Boolean circuit depth?

Disclaimer: I admit that the question is not very clear. I think it cannot be helped because the question is very open-ended. First of all, I present the interested type of circuits. We only consider ...
1
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0answers
97 views

Complexity of planted root of a system of quadratic homogeneous polynomials?

Given homogeneous degree $2$ randomly chosen polynomials $f_1,\dots,f_{m}$ in $\mathbb Z[x_1,\dots,x_n,y_1,\dots,y_n]$ each with only monomials $x_iy_j$ with condition that the system $f_1=\dots=f_{m}=...
1
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0answers
19 views

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 ...
1
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1answer
101 views

Optimal evaluation of polynomials / rational functions

A common way to compute the value a polynomial is to write it in Horner form. However, this isn't always the fastest way to evaluate it. Setting aside concerns of numerical precision, take the ...
8
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1answer
201 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 ...
5
votes
1answer
117 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 ...
2
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0answers
136 views

Complexity of solving a polynomial equation

Given a polynomial equation of degree n with m variables, that is guaranteed to have at least one solution, what is would be the ...
2
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0answers
55 views

Bit complexity of factoring univariate polynomial over $\mathbb{Q}$ (rationals)

What is the bit complexity of finding all the irreducible factors $f_1, ..., f_r$ of a degree-$d$ polynomial $f(x) = \sum_{i=0}^d a_i\cdot x^i \in \mathbb{Q}[x]$ whose all coefficients are $B$-bit ...
23
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3answers
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 = 1}^n\left(1-x_i\...
4
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1answer
279 views

If a root||nonce Proof-of-Work certificate is prime, can it be used in any other interesting proofs?

Because Bitcoin and many other cryptocurrency mining certificates are "rare" in that their respective hash is less than a very small number, can we leverage their rarity in probabilistic proofs of ...
6
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2answers
572 views

a polynomial representation of boolean functions

I came up with this linear transformation to map boolean functions to polynomials and it seems to have some nice properties. I was wondering if there is any reference describing this (and/or similar) ...
1
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0answers
56 views

Does deterministic PIT produce deterministic irreducible polynomial generation?

In $\Bbb F_q[x]$ given $d\in\Bbb N$ there is a deterministic $O(poly(nd\log q))$ algorithm to find an irreducible polynomial with $d=deg(x)$ under $GRH$ and an unconditional randomized algorithm. Do ...
11
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1answer
289 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-...
2
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0answers
98 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 ...
9
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0answers
283 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 ...
1
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0answers
91 views

Deciding reachability under iterated independent polynomial mapping

For any $1\leq i\leq m$, $f_i: \mathbb{Q}\rightarrow \mathbb{Q}$ is a polynomial mapping over $x_i$, where $\mathbb{Q}$ is the set of rationals. For $\vec{a}_0=(a_1, \cdots, a_m)\in \mathbb{Q}^m$, we ...
2
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0answers
186 views

Proof of a (simple) lemma by Aaronson

I am reading this article, and I need help with an apparently obvious proof. The lemma (on page 5), that I want to know the proof of, is this: Let $p : \{0,1\}^N \rightarrow \mathbb{R}$ be a real ...
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0answers
267 views

Computational complexity of polynomial interpolation with k non-zero terms

I am attempting to find a complexity for computing the order polynomial of partially ordered sets on a special family, and have come across the following problem. Assume we have the following values ...
3
votes
0answers
44 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 ...
3
votes
1answer
352 views

Factoring with LLL when the form of the factors is given

Given a degree $2k$ reducible polynomial $$f(x)=\sum_{i=0}^{2k}a_ix^i\in\Bbb Z[x]$$ with $$\text{gcd}(a_{2k},\dots,a_0)=1$$ that is known to be of the form $f_1(x)f_2(x)$ with $\text{deg}\big(...
8
votes
0answers
173 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+...
24
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1answer
747 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 ...
3
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2answers
197 views

Hardness of finding roots of a degree $2$ polynomials over $\mathbb{F}_2$

Since every $3$-SAT instance with $n$ variables can be expressed as a degree-$3$ polynomial over $\mathbb{F}_2$ with $n$ unknowns, the NP-hardness of $3$-SAT directly translates to NP-hardness of ...
4
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0answers
194 views

Circuit complexity class of polynomial factoring and Hensel lifting in Zassenhaus' algorithm?

Given a primitive polynomial (gcd of coefficients is $1$) in $\Bbb Z[x]$ we have a polynomial time factoring algorithm for this that runs in time polynomial in degree $d$ and number of bits in ...
6
votes
1answer
228 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 ...
5
votes
1answer
188 views

Finding a positive point for a collection of polynomials

I am wondering about the complexity of the following problem: Given $k$ polynomials $p_1(x_1, \ldots, x_n)$, $p_2(x_1, \ldots, x_n)$, $\ldots$, $p_k(x_1, \ldots, x_n)$ over the $n$ real ...
10
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2answers
145 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 ...
3
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0answers
74 views

Finding degree two subfield

Let $K=\frac{\mathbb{Q}[x]}{<f(x)>}$ where $f(x)$ is irreducible over $\mathbb{Q}$ and has even degree. I want to find $K_2$ such that $ \mathbb{Q} \subseteq K_2\subseteq K$ and $[K_2:\mathbb{Q}]...
5
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0answers
109 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....
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0answers
73 views

Approximating circuits with polynomial of low degree, can't understand small detail in the proof

I'm looking at the proof of this lemma: Lemma For every integer $t>0$, there exists a (proper) polynomial of total degree $(2t)^d$ that differs with $C$ on at most $size(C) 2^{n-t}$ inputs Where ...
0
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0answers
59 views

Identity testing of margins of boolean functions

Consider a boolean function $f(\vec{x},\vec{y})$, we define an (integer-valued) function $g(\vec{x})=\sum_{\vec{y}}f(\vec{x},\vec{y})$, i.e., $g$ can be considered as a (sort of) margin of $f$. The ...
1
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1answer
247 views

Existence of solution for a system of multi-variate polynomial equations and in-equations

Formally, I have 2 finite sets of polynomials : $P = \{p_1, p_2, p_3, ...p_m\}$ and $Q = \{q_1, q_2, q_3, ...q_n\}$, where for $1 \leq i \leq m$, and $1 \leq j \leq n$, I have $p_i, q_j \in \mathbb{C}...
4
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1answer
149 views

A special case of the boolean multivariate quadratic polynomial problem

It's well known that in the general case, the boolean MQ problem: given $(f_1, \ldots, f_n) \in \mathbb{F}_2[x_1, \ldots, x_m]$ with $\deg(f_i) = 2$, can we find a solution $\vec{y}: f_i(\vec{y}) = 0$...
6
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2answers
251 views

How to find a non-zero point of a non-zero polynomial of low degree?

Given a circuit that computes a polynomial $P(x_1 \dots x_n)$ of low formal degree over some large field $\mathbb{F}$. Moreover, given a point $X \in \mathbb{F}^n$, such that $P(X) \neq 0$. Can one ...
10
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1answer
250 views

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 ...
28
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6answers
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 ...
5
votes
1answer
299 views

Complexity of multi-linear polynomial computing Boolean function

Let $f:\{0,1\}^{n}\longmapsto\{0,1\}$ be a Boolean function. As usual, let $C(f)$ denote circuit complexity of $f$, i.e, the size of the smallest Boolean circuit computing $f$. As we know that every ...
40
votes
12answers
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 ...
1
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0answers
60 views

The curve used in Parvaresh-Vardy decoding

Consider the Parvaresh-Vardy list decoder. As I understand it, the idea is to decide on a curve over an extension field of the form $(f,f^h mod E, f^{h^2} mod E,\dots)$ and then evaluate each of ...
0
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0answers
65 views

Complexity of variant polynomial factorization

Few years back a question on whether a variant of integer factorization was $\mathsf{NP}$ complete was asked, which according to state of mathematics today does not have a conclusive answer. Is ...
1
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0answers
153 views

Algorithm for multiplying multivariate polynomials in a commutative ring

Let $R$ be a commutative ring. Let $f(x_1, \dots, x_n), g(x_1, \dots, x_n)$ be two multivariate polynomials with the same $x_i$-terms with maximal total degree $\delta$, but with different ...
1
vote
2answers
266 views

Does approximation degree of AND depend on error?

Denote $\underline{x}\triangleq x_1,\dots,x_n$. Given a boolean function $f:\{0,1\}^n\rightarrow\{0,1\}$, let $p_\epsilon(\underline{x})\in\Bbb R[\underline{x}]$ be minimum multilinear multivariate ...
1
vote
1answer
178 views

Classes of boolean functions where reasonable lower bounds on approximate degree is unknown?

Let $\underline{x}\triangleq x_1,\dots,x_n$. Given a Boolean function $f:\{0,1\}^n\to\{0,1\}$, we say that $p(\underline{x})\in\Bbb{R}[\underline{x}]$ is an $\epsilon$-approximation to $f$ if for ...
7
votes
1answer
211 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 ...