# Questions tagged [polynomials]

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

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### Characteristic polynomial

The characteristic polynomial of a square matrix A is defined as the polynomial pA(x) = det(Ix-A) where I is the identity matrix and det the determinant. Note that this definition always gives us a ...
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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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### 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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### 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)...
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### Find the coefficients of a rational generating function

If we write a sequence of numbers as the coefficients of a power series, then that power series is called the (ordinary) generating function (or G.f.) of that sequence. That is, if for some function <...
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### Add up two algebraic numbers

Definitions An algebraic number is a number that is a zero of a non-zero polynomial with integer coefficients. For example, the square root of 2 is algebraic, ...
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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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### Find the polynomial

We know that f is a polynomial with non-negative integer coefficients. Given f(1) and f(1+f(1)) return f. You may output f as a list of coefficients, an ASCII formatted polynomial, or similar. ...
337 views

### Detect a Symmetric polynomial [closed]

A symmetric polynomial is a polynomial which is unchanged under permutation of its variables. In other words, a polynomial f(x,y) is symmetric if and only if ...
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### Calculate Power Series Coefficients

Given a polynomial p(x) with integral coefficients and a constant term of p(0) = 1 or -1, and a nonnegative integer ...
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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. The Sidi polynomials have several interesting properties, but so do their coefficients. The ...
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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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### One out of Infinity: Interpolating polynomials [duplicate]

For this challenge, when given a list of (x,y) points your submission needs to output a polynomial function that goes through all of them. For example, if your points were ...
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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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### Shamir's Secret Sharing

Given n (the number of players), t (the threshold value), and s (the secret), output the <...
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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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### 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, ...
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### Polynomialception

Given two polynomials f,g of arbitrary degree over the integers, your program/function should evaluate the first polynomial in the second polynomial. ...
828 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 ...
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### Is it a factor of a polynomial?

A polynomial is divisible by a factor (x-n) if f(n)=0 for a function f. Your job: to ...
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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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### 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 ...
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### Laplace transform of a polynomial

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 the integral from 0 to ...
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### 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 ...