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# Questions tagged [polynomials]

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

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### Is this polynomial a square?

Given an integral polynomial $p$, determine if $p$ is a square of another integral polynomial. An integral polynomial is a polynomial with only integers as coefficients. For example, $x^2+2x+1$ ...
676 views

### Chromatic polynomial of a graph

Given a undirected graph $G$ and a integer $k$, how many $k$-coloring does the graph have? Here by a $k$-coloring, we mean assigning one of the $k$ colors to each vertex of the graph, such ...
140 views

### Order of an algebraic number

Consider some arbitrary polynomial with integer coefficients, $$a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0$$ We'll assume that $a_n \ne 0$ and $a_0 \ne 0$. The solutions to this polynomial ...
1k views

### Are you a probabilist or a physicist?

Hermite polynomials refer to two sequences of polynomials: The "probabilist's Hermite polynomials", given by $${He}_n(x) = (-1)^n e ^ \frac {x^2} 2 \frac {d^n} {dx^n} e ^ {-\frac {x^2} 2}$$ ...
1k views

532 views

### Golfing Expressions

We can write mathematical expressions using the standard math operators (,),+,...
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 ...
953 views

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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 ...
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)...
416 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 <...
441 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, ...
948 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 ...
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 ...
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. ...
342 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 ...
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 &...
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 ...