# Questions tagged [polynomials]

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

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 <...
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### 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., "...
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 ...
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 ...
877 views

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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Self Referential Polynomials

For every given degree n it is possible to construct (at least one) an integral polynomial p such that ...
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### 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(...
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### 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 ...
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### 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 ...
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### Polynomial -> Integrate

Given a polynomial in one variable with rational coefficients, output an equivalent expression containing only 1, variables, and definite integrals. For example, -...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Locally invert a Polynomial

Challenge Given a polynomial p with real coefficients of order 1 and degree n, find another ...
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### 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 ...
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### 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 ...