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1 vote
2 answers
66 views

Subquadratic multiplication of polynomials in the max-plus/tropical semiring

Is there an algorithm to multiply two polynomials with coefficients in the max-plus semiring $(\mathbb{Z}\cup\{-\infty\}, \max, +)$ which is faster than the trivial one? I'm interested in the ...
2 votes
1 answer
99 views

NP-hardness of solving systems of *homogeneous* polynomial equations

It is well-known that deciding if a system of quadratic polynomial equations in several variables admits a solution in a finite field is NP-complete. There is a simple reduction from 3SAT, that works ...
1 vote
1 answer
34 views

Distinction between square roots in cyclic fields

Let $\mathbb{F}=\mathbb{Z}/p\mathbb{Z}$ a cyclic field. Where $p$ is fixed Let $(H)_{n\in\mathbb{N}} \in \mathbb{Z}[x_1,\dots]^{\mathbb{N}}$ a family of polynomials with $H_n\in \mathbb{Z}[x_1,\dots,...
0 votes
0 answers
24 views

Multiplying two bivariate polynomials using FFT for univariate polynomials multiplication

Let $$f(x,y)=\sum_{0\le i\le n , 0 \le j \le d}a_{i,j}x^iy^j$$ $$g(x,y)=\sum_{0\le i\le n , 0 \le j \le d}b_{i,j}x^iy^j$$ We want to multiply $f g$. I did the following: $$f(x,x^{2n+1})=\sum_{0\le i\...
6 votes
4 answers
31k views

Converting Polynomials into Binary form

How can a polynomial such as $x^3 + 1$ be converted to its binary form of $1001$. Likewise, $10100001 = x^7 + x^5 + 1.$
2 votes
2 answers
2k views

Find all rational roots of a polynomial equation

I'm going to try to design an algorithm to find all the rational roots of a polynomial equation in range [a, b]. Can someone please tell me which algorithm currently solves the problem with lowest ...
2 votes
1 answer
168 views

Does the reliability of polynomial hashing depend on whether the modulus is prime, for coprime base and modulus?

A polynomial hash of a string $s$ with base $b$ and modulus $M$ is defined as $$ H(s) = (s_0 + s_1 b + s_2 b^2 + \dots + s_{n-1} b^{n-1}) \mod M. $$ I have proven (and this is quite obvious) that ...
1 vote
1 answer
22 views

Best internal representation of a random variable to enable iterative sampling and interpolation/regression

Let $[0,100]$ denote the interval of real numbers between $0$ and $100$. Given a function $f:[0,100]^n \rightarrow \mathbb{R}^+$, I want to implement the following simple algorithm to search for the ...
1 vote
0 answers
88 views

What is the difference between $O$ and $\widetilde{O}$?

We know that $\widetilde{O}(f(n))$ — $O$ with a tilde above it — which means $O(f(n) \text {polylog}(f(n)))$, i.e., $O(f(n) (\log f(n))^k)$ for some $k$. Also I have seen in Wikipedia that $n2^n=\...
0 votes
1 answer
69 views

Algorithm for determining an algebraic positivity property of a multivariate polynomial

I am considering multivariate polynomials with integer coefficients that can be expressed as sums of products of terms of the form $y_i-z_j$ for positive integers $i$ and $j$. I am trying to find an ...
0 votes
1 answer
55 views

Sumcheck protocol - how are these 2 polynomials different?

I am going through Justin Thaler's book - https://people.cs.georgetown.edu/jthaler/ProofsArgsAndZK.pdf - "Proofs, Arguments, and Zero-Knowledge" He is describing the Sumcheck Protocol on ...
2 votes
0 answers
135 views

Optimal reassociations is NP-hard?

Consider signed integers with common addition and multiplication. Reassociation of expression is another equivalent form. Say expressions: ...
0 votes
1 answer
38 views

Which one grows faster, an exponential function where the exponent grows faster than logarithmic or a factorial powered by n?

Which function grows faster: $$f(n) = 4^{n^2 \log_2 n} \text{ or } g(n) = (n!)^n$$ Which is true? $f(n) = O(g(n))$ $g(n) = O(f(n))$ i.e., $f(n) = \Theta(g(n))$ none of the above? For lower values of ...
0 votes
0 answers
49 views

Progress towards a Polynomial time factoring algorithm?

This is probably insignificant, but I was messing around with polynomials, and found out that, if we have a number, n = pq that we want to factor, if we expand (k+1)^n -k^n - 1, mod n, the first ...
0 votes
2 answers
84 views

Polynomial representations of Boolean functions

The AND boolean function $AND(x)$ can be represented using the polynomial $P(x) = x_1x_2\cdots x_n$. I have a few questions: Is there a similar polynomial for the PARITY boolean function? Is there a ...
1 vote
1 answer
63 views

product of every difference

Given a sorted array where every element is distinct, we need to evaluate product of every difference, modulo $ 10^9 + 7 $ $$ \prod_{i < j} (arr[j] - arr[i]) \% (10^9 + 7) $$ Best approach I can ...
2 votes
0 answers
56 views

Algorithm to compute sum of quotient polynomials

Let $f(X)$ be a polynomial in $\mathbb{F}_p[X]$ for some prime $p$ (of size 256 bits) that is not necessarily FFT-friendly. Let $a_1,\cdots,a_n$, $b_1,\cdots, b_n$ be $\mathbb{F}_p$ elements. What is ...
7 votes
2 answers
389 views

Efficient algorithm to translate/shift polynomials

I have a polynomial $P(x)$, and given some constant $d$, I need to find the polynomial $P(x+d)$. For example, if $P(x)=x^2$ and $d=1$, then the result would be $P(x+1)=(x+1)^2=x^2+2x+1$ (with the ...
0 votes
1 answer
77 views

O(nlogn)-time complexity

Is there a $O(nlogn)$ time algorithm for computing $p(x)=\sum\limits_{i=0}^na_ix^i$ ? I think with the method below I get O(n), but I need O(nlogn) Hint: there's a way to calculate $x^i$ more ...
1 vote
1 answer
437 views

CRC computation speed vs polynomials features

I tried to find information about how features of a CRC polynomials influence computation speed of implementations. It is obvious that (depending from the CPU architecture the algorithm runs on) ...
1 vote
0 answers
47 views

What algorithms apart from FFT get a computational boost by leveraging complex numbers?

If we think of the algorithmic problem of doing a convolution of two arrays, it turns out that converting them to the frequency domain first and then doing an element-wise product is equivalent. And ...
0 votes
1 answer
50 views

Can an arithmetic circuit have multiple outputs?

An arithmetic circuit relates to calculating the value of a polynomial given some inputs. But is it still considered a circuit if the DAG corresponds to the evaluation of multiple polynomials that ...
2 votes
0 answers
82 views

Fastest algorithm for polynomial multiplication in 256-bit finite fields

I am looking for the fastest algorithm (in practice) to multiply two polynomials $f(X)\cdot h(X)$ in $\mathbb{F}_p[X]$. The prime $p$ is roughly $256$ bits but the integer $p-1$ might not have any ...
1 vote
1 answer
102 views

Complexity of multiplying bivariate polynomials of degree n

Let $P(X,Y)$ and $Q(X,Y)$ be two bivariate polynomials of degree at most $n$. Using $O(n^2)$ FFTs, we can compute the product $PQ$ in time $O(n^3\log n)$. Q: Is there a faster algorithm to compute $PQ$...
1 vote
0 answers
25 views

Time and Space Complexity of Isolating the Roots of a Polynomial

Can someone provide a list of references of papers, books, etc. presenting results on the time and space complexity of the problem of isolating the roots of a univariate polynomial $P(t)$ with integer ...
1 vote
0 answers
169 views

What's the fastest algorithm for polynomial interpolation in finite field with prime order at points 1, 2, 3, ..., n?

Suppose we have finite field $\mathbb{F}_p$ with prime order $p$. Can we find polynomial in $\mathbb{F}_p[x]$ having given values $a_1, a_2, ... a_n$ at $x=1, 2, ..., n$ in less than $O(n^2)$ ...
4 votes
0 answers
60 views

Polynomial multiplication in finite field without smooth-order roots of unity

I am working in a finite prime field $\mathbb{F}_p$ that does not have primitive $n$-th roots of unity for any large smooth integer $n$, which makes FFTs a bit difficult. If I need to compute a ...
2 votes
3 answers
631 views

Deciding whether an integer polynomial has an integer root

This is a question written by my instructor Z. Loria . Consider the following problem: Given a polynomial $p(x) = \sum_{i=0}^n a_ix^i$, where $a_i$ are integers, is there a natural number $n \in \...
0 votes
1 answer
132 views

Finding $Aeven(x)$ and $Aodd(x)$ for a Fast Fourier Transform (FFT) problem?

In this article about FFT the author used FFT on this polynomial. $A(x) = 3+2x+3x^2+4x^3$ Using $ A(x)= Aeven(x^2) + xAodd(x^2)$ the author determined the following for $Aeven(x)$ and $Aodd(x)$ $ ...
2 votes
1 answer
51 views

Is there a simplistic way of describing the proof to the undecidability of David Hilbert's 10th problem?

I recently have been reading a bunch about David Hilbert's famous 10th problem, and trying to understand its proof. I am currently in the process of reading through an explanation of the proof, given ...
1 vote
0 answers
48 views

Efficient bit-level implementation of Boolean polynomials with few variables

A Boolean polynomial in $n$ variables $x_1, \dots, x_n$ is an expression of the form $$\sum_{\mathbf{s} \in \{0,1\}^n} c_{\mathbf{s}} x_1^{s_1} \cdots x_n^{s_n}, \quad \text{ where } c_s \in \{0,1\} .$...
15 votes
2 answers
4k views

What is the most efficient algorithm to compute polynomial coefficients from its roots?

Given $n$ roots, $x_1, x_2, \dotsc, x_n$, the corresponding monic polynomial is $$y = (x-x_1)(x-x_2)\dotsm(x-x_n) = \prod_{i}^n (x - x_i)$$ To get the coefficients, i.e., $y = \sum_{i}^n a_i x^i$, a ...
0 votes
0 answers
42 views

From roots to coefficients of a polynomial [duplicate]

Polynomials are usually written as a sum of powers (or various products of generators) and Google gives me lots of results on how to get from that to the form that is a product of degree-$1$ ...
-1 votes
1 answer
22 views

Given x find a polynomial such that pol(x)=a for a known a?

You are given x,a. Find a polynomial p(y) with the leading coafficent 1 such that p(x)=a. How to write an algorithem to solve this efficently? I have no idea where to start
1 vote
0 answers
62 views

Finding points of local maximum error in Remez algorithm

So the Remez algorithm is an algorithm for finding optimal polynomial approximations or at the very least for converging towards them. To find an approximation of $f$ by an $N$th degree polynomial the ...
3 votes
1 answer
198 views

Computing coefficients of $p(x)^n$ in time $O(n \log n)$

For homework I've to give an algorithm that computes the coefficients of the polynomial $p(x)^n$ in time $O(n\log n)$, where $p(x)$ is a polynomial of degree 7. As an hint I'm told to consider first ...
0 votes
0 answers
53 views

Time to evaluate a product of binomials $1 + x^i$

I am considering the asymptotic analysis required to convert a polynomial of the shape $$P(x) = \prod_{i = 1}^{n}(1 + x^{s_i})$$ to its "full" representation, for example $$P(x) = 1+x^3+x^5+...
0 votes
1 answer
314 views

Multiplication of polynomials in value representation as done for Fast Fourier Transform

I am trying to understand the discrete Fast Fourier Transform. I get the idea of switching between coefficient and value representations to and then back but I am stuck in figuring out how the ...
1 vote
0 answers
96 views

Optimize binary multivariate polynomial multiplication

You have a few variables that can only assume the values 0 or 1 and you use those to form two polynomials. Is there a way to multiply the two polynomials that is faster than O(n²) where n is the ...
1 vote
1 answer
85 views

How polynomial interpolation of polynomial multiplication algorithm works?

I'm trying to apply the following algorithm from DPV's textbook to an example to see how it works. First, the algorithm is as following: Algorithm [DPV, p. 60] Input: Coefficients of two polynomials. ...
1 vote
0 answers
68 views

Express polynomial as sum of two lower-degree polynomials, modulo another

Suppose I have a polynomial $p(x)$, and a "modulus" polynomial $q(x)$ of degree $d$. I want to find two polynomials $r_1(x),r_2(x)$ of degree $\le d_1,d_2$ such that $$p(x) \equiv r_1(x) x^...
2 votes
0 answers
120 views

Decide whether a polynomial has a root

Let $A$ be a ring such that all elements of $A$ are complex computable numbers. I'm interested in knowing whether the decision problem that asks, given $P\in A[X]$, if $P$ has a root in $A$ is ...
1 vote
2 answers
66 views

Show that the OR of n variables cannot be expressed as a polynomial over Fp of degree less than n

Here is a question from Computational Complexity by Arora and Barak: Show that representing OR of $n$ variables $x_1,x_2,\dots,x_n$ exactly over a polynomial in $GF(q)$ requires degree exactly $n$. (...
1 vote
0 answers
37 views

Binomial basis and the usual basis of polynomial algebra $\mathbb C(X)$

Consider the polynomial algebra $\mathbb C[X]$. Then the set $\{1, X, X^2,\dots,\}$ forms a vector space basis for this algebra. In general, we know that the set $\{P_n(X) \in \mathbb C[X]: n \ge 0 \...
0 votes
1 answer
51 views

Represent a DNF formula as a multivariate linear formula?

Lets say I have the following DNF: (x or y) and (z or i) / $(x\lor y)\land(z\lor i)$ How do I convert that into a polynomial form?
1 vote
0 answers
45 views

How do I decode a received polynomial code with an error?

As a message I get (5,0,1,3), which is coding a sequence of numbers of length 2 in $\mathbb{F}_7$ as polynom with the 4 support points a1 = 0, a2 = 1, a3 = 2, a4 = 6. In the transimission occured an ...
1 vote
1 answer
194 views

Given an array $a$, we have to find product of $a_{j}$-$a_{i}$ modulo $998244353$ over all $i$ and $j$ given $j>i$

Given an array $a$, we have to find product of $a_{j}$-$a_{i}$ modulo $998244353$ over all $i$ and $j$ given $j>i$. For eg. Let the array be $1,2,3$ then my answer will be calculated as- $(2-1)$.$(...
3 votes
1 answer
233 views

Is deciding solvability of systems of quadratic equations with integer coefficients over the reals in NP?

In the book 'Computational Complexity' by Arora and Barak the following question is posed (exercise 2.20.): Let REALQUADEQ be the language of all satisfiable sets of quadratic equations over real ...
2 votes
0 answers
219 views

Understanding CRC Computation with PCLMULQDQ

I am currently reading this paper which shows how to calculate CRC using the instruction PCLMULQDQ. I don't quite understand the equations in it yet. Starting with this one for the definition of ...
1 vote
0 answers
64 views

How to compute the first n coefficients using number theoretic transform (NTT)?

I need to find the first $n$ coefficients of $$\prod_{i = 1}^{i = q}(1 + x^{a_i})^{b_i}$$ modulo a NTT favourable prime. Can someone suggest an algorithm with worst-case complexity $O(n\log n)$ or $O(...