# Questions tagged [polynomials]

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### Computational complexity of factoring univariate polynomials with positive integer coefficients

I am interested in the computational complexity of the following problem. Input: a polynomial p(x) with positive integer coefficients Output: a factorization of p(x) into irreducible factors having ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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+...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 $[K_2:\mathbb{Q}]... 0answers 111 views ### Algebraic dependence of roots of irreducibles over a finite field I asked this question in Math SE too, but I have since modified it to make it more suited here. Also, in hindsight, the question itself was more algorithmic and was a better fit here. https://math.... 0answers 75 views ### Approximating circuits with polynomial of low degree, can't understand small detail in the proof I'm looking at the proof of this lemma: Lemma For every integer$t>0$, there exists a (proper) polynomial of total degree$(2t)^d$that differs with$C$on at most$size(C) 2^{n-t}$inputs Where ... 0answers 63 views ### Identity testing of margins of boolean functions Consider a boolean function$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$. The ... 1answer 514 views ### Existence of solution for a system of multi-variate polynomial equations and in-equations Formally, I have 2 finite sets of polynomials :$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}...
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 $\vec{y}: f_i(\vec{y}) = 0$...