Questions tagged [polynomials]

For challenges involving polynomials, mathematical expressions that consist of variables and coefficients.

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23
votes
5answers
483 views

Recover polynomial \$f(x)\$ from \$f^2(x)\$

Related: Calculate \$f^n(x)\$, Polynomialception Challenge Given a polynomial \$f(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_k x^k\$ of order \$k\$, we can compute its composition with itself \$f\left(f(...
20
votes
25answers
1k views

Calculate the n-th iterate of a polynomial for a specific value; fⁿ(x)

Given a polynomial function f (e.g. as a list p of real coefficients in ascending or descending order), a non-negative integer n, and a real value x, return:    f n(x) i.e. the value of f (f (f (…f (x)...
27
votes
19answers
3k views

Calculate the Ultraradical

What is the Ultraradical? The ultraradical, or the Bring radical, of a real number \$a\$ is defined as the only real root of the quintic equation \$x^5+x+a=0\$. Here we use \$\text{UR}(\cdot)\$ to ...
10
votes
0answers
210 views

Golfing Expressions

We can write mathematical expressions using the standard math operators (,),+,...
18
votes
18answers
2k views

Determine the degree of a polynomial

Background: For this challenge, a polynomial looks like this: $$P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_2x^2+a_1x+a_0$$ The degree, \$n\$, is the highest power \$x\$ is raised to. An example of a degree 7 ...
12
votes
12answers
864 views

Definite integral of polynomial functions

You will need to evaluate the definite integral (bounded by \$a\$ and \$b\$) of a certain polynomial function that takes the form of: $$\int_a^b \left( k_n x^n + k_{n-1} x^{n-1} + \cdots + k_2x^2 + ...
23
votes
28answers
2k views

“Factorise” a quadratic

When learning to factorise quadratics in the form \$x^2 + ax + b\$, a common technique is to find two numbers, \$p, q\$ such that $$pq = b \\ p + q = a$$ as, for such numbers, \$x^2 + ax + b = (x + p)(...
28
votes
16answers
3k views

Многочлены Чебышёва (Chebyshev Polynomials)

Chebyshev Polynomials are a family of orthogonal polynomials that pop up in all kinds of places in math, and they have a lot of quite interesting properties. One characterization of them is that they ...
12
votes
18answers
1k views

Polynomial Laplace transform

This is a repost of this challenge, intended to revamp it for looser I/O formats and updated rules You are to write a program which takes an integer polynomial in \$t\$ as input and outputs the ...
13
votes
3answers
948 views

Irreducible polynomials over GF(5)

A polynomial with coefficients in some field F is called irreducible over F if it cannot be decomposed into the product of lower degree polynomials with coefficients in F. Consider polynomials over ...
30
votes
13answers
4k views

Absolute Sums of Sidi Polynomial Coefficients

Background The Sidi polynomial of degree \$n\$ – or the \$(n + 1)\$th Sidi polynomial – is defined as follows. $$S_n(x) = \sum^n_{k=0}s_{n;k}x^n \text{ where } s_{n;k} = (-1)^k\binom n k (k+1)^n$$ The ...
14
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10answers
3k views

Find The Local Maxima And Minima

Definition The maxima and minima of a given function are the largest and smallest values of the function either within a given range or otherwise within the entire domain of the function. Challenge ...
8
votes
2answers
599 views

Laplace transform of a polynomial [duplicate]

Your goal is to write a program that will print out the Laplace transform of a polynomial function with integer coefficients \$f(x)\$. The Laplace transform of \$f(x)\$ is defined as \$\int_0^\infty f(...
15
votes
5answers
2k views

Polynomial Long Division

Implement polynomial long division, an algorithm that divides two polynomials and gets the quotient and remainder: (12x^3 - 5x^2 + 3x - 1) / (x^2 - 5) = 12x - 5 R 63x - 26 In your programs, you will ...
-2
votes
1answer
135 views

The Perfect Polynomial [closed]

The coefficients of a perfect square polynomial can be calculated by the formula \$(ax)^2 + 2abx + b^2\$, where both a and b are integers. The objective of this challenge is to create a program that ...
13
votes
9answers
1k views

Characteristic polynomial

The characteristic polynomial of a square matrix \$A\$ is defined as the polynomial \$p_A(x) = \det(Ix-A)\$ where \$I\$ is the identity matrix and \$\det\$ the determinant. Note that this definition ...
12
votes
6answers
511 views

Multiply numerical polynomials

A numerical polynomial is a polynomial \$p\$ in one variable with rational coefficients such that for every integer \$i\$, \$p(i)\$ is also an integer. The numerical polynomials have a basis given by ...
24
votes
16answers
2k views

Laguerre Polynomials

Laguerre polynomials are solutions to Laguerre's equation, a second-order linear differential equation: \$xy''+(1-x)y'+ny=0\$. For a given value of n, the solution, y, is named \$L_n(x)\$. The ...
24
votes
9answers
1k views

Calculate Power Series Coefficients

Given a polynomial \$p(x)\$ with integral coefficients and a constant term of \$p(0) = \pm 1\$, and a non-negative integer \$N\$, return the \$N\$-th coefficient of the power series (sometimes called &...
21
votes
12answers
2k views

Symbolic Differentiation of Polynomials

Symbolic Differentiation 1: Gone Coefishin' Task Write a program that takes in a polynomial in x from stdin (1 < deg(p) < 128) and differentiates it. The input polynomial will be a string of ...
15
votes
14answers
1k views

Generating generating expressions for sequences

(yes, "generating generating" in the title is correct :) ) Context In middle (?) school we are taught about sequences and, in particular, we are taught about linear sequences where the ...
20
votes
8answers
1k views

Counting Distinct Real Roots of Low-Degree Polynomials

Challenge: I want to know about the real roots of polynomials. As a pure mathematician, I care about the existence of such roots, rather than their numeric values. The challenge is to write the ...
9
votes
5answers
506 views

Computing a specific coefficient in a quotient of polynomials

Context After "Computing a specific coefficient in a product of polynomials", asking you to compute a specific coefficient of polynomial multiplication, I wish to create a "mirror" challenge, asking ...
12
votes
5answers
1k views

Polynomial Interpolation

Write a program that performs Polynomial Interpolation using true arbitrary precision rational numbers. The input looks like this: f(1) = 2/3 f(2) = 4/5 f(3) = 6/7 ... You may assume that there's ...
15
votes
5answers
912 views

Define the finite field GF(9)

\$GF(9)\$ or \$GF(3^2)\$ is the smallest finite field whose order isn't a prime or a power of two. Finite fields of prime order aren't particurlarly interesting and there are already challenges for \$...
14
votes
5answers
809 views

Algebraic curve plotter

An algebraic curve is a certain "1D subset" of the "2D-plane" that can be described as set of zeros {(x,y) in R^2 : f(x,y)=0 }of a polynomial <...
10
votes
2answers
890 views

Find the largest root of a polynomial with a neural network

The challenge Find the smallest feedforward neural network such that, given any 3-dimensional input vector \$(a,b,c)\$ with integer entries in \$[-10,10]\$, the network outputs the largest (i.e., "...
21
votes
4answers
487 views

Compute height of Bowl Pile

Bowl Pile Height The goal of this puzzle is to compute the height of a stack of bowls. A bowl is defined to be a radially symmetric device without thickness. Its silhouette shape is an even ...
11
votes
2answers
426 views

​Plane​ ​Blow​up​

The Blow-up is a powerful tool in algebraic geometry. It allows the removal of singularities from algebraic sets while preserving the rest of their structure. If you're not familiar with any of that ...
5
votes
2answers
877 views

Point-free madness

This challenge is about Haskell point-free style polynomial functions. Although you don't need to know Haskell language to do this challenge, Haskellers might have an advantage here. Point-free ...
19
votes
17answers
2k views

Find Integral Roots of A Polynomial

Challenge The challenge is to write a program that takes the coefficients of any n-degree polynomial equation as input and returns the integral values of x for which the equation holds true. The ...
29
votes
26answers
3k views

Fundamental Solution of the Pell Equation

Given some positive integer \$n\$ that is not a square, find the fundamental solution \$(x,y)\$ of the associated Pell equation $$x^2 - n\cdot y^2 = 1$$ Details The fundamental \$(x,y)\$ is a pair of ...
13
votes
6answers
1k views

Ryley's Theorem

S. Ryley proved following theorem in 1825: Every rational number can be expressed as a sum of three rational cubes. Challenge Given some rational number \$r \in \mathbb Q \$ find three rational ...
12
votes
3answers
257 views

Self Referential Polynomials

For every given degree n it is possible to construct (at least one) an integral polynomial p such that ...
17
votes
11answers
2k views

Cyclotomic polynomial

Background (skip to definitions) Euler proved a beautiful theorem about the complex numbers: eix = cos(x) + i sin(x). This makes de Moivre's theorem easy to prove: (eix)n = ei(nx) (cos(x) + i sin(...
18
votes
19answers
3k views

Evaluate polynomial expression string

Create a function which takes a polynomial equation, a value for x and returns the result of the operation. Example: given ...
25
votes
1answer
1k views

Find real roots of a polynomial

Write a self-contained program which when given a polynomial and a bound will find all real roots of that polynomial to an absolute error not exceeding the bound. Constraints I know that Mathematica ...
11
votes
3answers
473 views

Polynomial -> Integrate

Given a polynomial in one variable with rational coefficients, output an equivalent expression containing only 1, variables, and definite integrals. For example, -...
3
votes
2answers
79 views

Generate lowest degree polynomial from sequence [duplicate]

Introduction A sequence of numbers is passed in as the input. The program has to generate the lowest degree polynomial possible. This was my first programming project in college and it would be ...
38
votes
23answers
3k views

Pretty Print Polynomials

Introduction Humans are a remarkable species, but we can be very awkward to understand sometimes—especially for computers. In particular, we seem to like writing polynomials in a very ...
14
votes
12answers
882 views

Construct a companion matrix

You have a number of polynomials who are lonely, so make them some companions (who won’t threaten to stab)! For a polynomial of degree n, there is an ...
15
votes
14answers
820 views

Euler-Poincaré-Characteristic of Polyhedra

Given a triangulation of the surface of a polyhedron p, calculate its Euler-Poincaré-Characteristic χ(p) = V-E+F, where ...
12
votes
2answers
347 views

Decompose Polynomials

Given an integral polynomial of degree strictly greater than one, completely decompose it into a composition of integral polynomials of degree strictly greater than one. Details An integral ...
11
votes
10answers
935 views

Rotate the Roots

Given a nonzero polynomial with integer coefficients and roots that are on the imaginary and on the real line such that if a is a root then so is ...
24
votes
7answers
1k views

Find the binarray!

We define a binarray as an array satisfying the following properties: it's non-empty the first value is a 1 the last value is a ...
20
votes
4answers
443 views

Locally invert a Polynomial

Challenge Given a polynomial p with real coefficients of order 1 and degree n, find another ...
8
votes
6answers
310 views

Simplify and Take Partial Derivative to a Polynomial String

Introduction Write a program to calculate the partial derivative of a polynomial (possibly multivariate) with respect to a variable. Challenge Derivatives are very important mathematical tools that ...
7
votes
2answers
282 views

Rational Polynomial Interpolation

Explanation In this task you'll be given a set of N points (x1,y1),…,(xN,yN) with distinct xi values and your task is to interpolate a polynomial through these points. If you know what Lagrange ...
19
votes
14answers
2k views

Discrete Convolution or Polynomial Multiplication

Given two non empty lists of integers, your submission should calculate and return the discrete convolution of the two. Interestingly, if you consider the list elements as coefficients of polynomials, ...
7
votes
8answers
1k views

Construct a polynomial with given roots

The challenge is to write the shortest function/program to find a polynomial given the roots. The function or program should take valid input; that is, a list of integers representing the roots of the ...