Questions tagged [polynomials]

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

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17
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 ...
11
votes
12answers
855 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 + ...
11
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 ...
22
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)(...
-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 ...
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 ...
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
504 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 ...
14
votes
5answers
863 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 \$...
10
votes
2answers
887 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., "...
26
votes
18answers
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
2answers
422 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 ...
20
votes
4answers
485 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 ...
4
votes
2answers
865 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 ...
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
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 ...
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 ...
11
votes
3answers
470 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 ...
15
votes
14answers
817 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
329 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 ...
8
votes
6answers
309 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 ...
14
votes
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 ...
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 ...
7
votes
2answers
281 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 ...
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 ...
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(...
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 ...
19
votes
22answers
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)...
12
votes
6answers
391 views

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 <...
7
votes
4answers
413 views

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, ...
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 ...
21
votes
8answers
1k views

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. ...
8
votes
3answers
338 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 ...
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 &...
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 ...
20
votes
4answers
439 views

Locally invert a Polynomial

Challenge Given a polynomial p with real coefficients of order 1 and degree n, find another ...
3
votes
3answers
109 views

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 ...
14
votes
5answers
798 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 <...
17
votes
12answers
2k views

Shamir's Secret Sharing

Given n (the number of players), t (the threshold value), and s (the secret), output the <...
12
votes
3answers
253 views

Self Referential Polynomials

For every given degree n it is possible to construct (at least one) an integral polynomial p such that ...
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, ...
22
votes
12answers
2k views

Polynomialception

Given two polynomials f,g of arbitrary degree over the integers, your program/function should evaluate the first polynomial in the second polynomial. ...
13
votes
3answers
936 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 ...
12
votes
5answers
463 views

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 ...
14
votes
12answers
873 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 ...