# Questions tagged [polynomials]

Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.

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### Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...
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### A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself

Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...
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### What Are Some Naturally-Occurring High-Degree Polynomials?

To construct J. H. Conway's look-and-say sequence, begin by putting down a 1 as the first entry. The other entries are found by saying the previous entry aloud, and writing what you hear. ...
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### A family of polynomials whose zeros all lie on the unit circle

I had posted the following problem on stack exchange before. Suppose $\lambda$ is a real number in $\left( 0,1\right)$, and let $n$ be a positive integer. Prove that all the roots of the ...
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### Polynomials with the same values set on the unit circle

Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). ...
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### Integers not represented by $2 x^2 + x y + 3 y^2 + z^3 - z$

EDIT, 9 March 2014: when I asked this in 2010, I did not have the courage of my convictions, and so did not ask for an if and only if proof, as Kevin Buzzard quite properly pointed out. Such problems ...
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### Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable?

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$, decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective, respectively, injective? -- And ...
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The following problem arose in collaborative work with Subhro Ghosh: Question: To any polynomial $P_n(z)=\sum_{i=0}^n a_i z^i =a_n \prod_{i=1}^n(z-z_i)$, attach the empirical measure of zeros $L_n=n^{... 3answers 1k views ### Is$x^{2k+1} - 7x^2 + 1$irreducible? Question. Is the polynomial$x^{2k+1} - 7x^2 + 1$irreducible over$\mathbb{Q}$for every positive integer$k$? It is irreducible for all positive integers$k \leq 800$. 6answers 2k views ### Bass' stable range of$\mathbf Z[X]$Let$n$be a positive integer and$A$be a commutative ring. The ring$A$is said to be of Bass stable range$\mathrm{sr}(A)\leq n$if for$a, a_1, \dots, a_n \in A$one has the following implication: ... 3answers 2k views ### when is the power of a nonnegative polynomial a sum of squares? There are polynomials that are not sum of squares. For example Motzkin gave the example$x^4y^2+x^2y^4+z^6-3x^2y^2z^2$in 1967. Is there a real polynomial$f\in{\mathbb{R}}[x_1,\ldots,x_n]$in ... 1answer 1k views ### Polynomials non-negative on the integers Let$P$be a real polynomial of exact degree$2n$($n \geq 1$) whose zeros are real numbers and such that \begin{equation*} P(j) \geq 0 \quad \text{for any} \quad j \in \mathbb{Z}. \end{equation*} ... 12answers 5k views ### When does 'positive' imply 'sum of squares'? Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares? Example. A positive integer does not ... 1answer 1k views ### How many polynomial Morse functions on the sphere? Let$f$be a homogeneous polynomial of degree$d$in$n$variables. Restricted to the unit sphere$S^{n-1}$, it might or might not be a Morse function. If$f$is a Morse function of degree$1$, you ... 5answers 2k views ### Given a polynomial f, can there be more than one constant c such that every root of f(x)-c is repeated? The question Let$f$be a nonconstant polynomial over$\mathbb{C}$. Let's say that a point$c \in \mathbb{C}$is unusual for$f$if every root$x$of$f(x) - c$is repeated. Can$f$have more than ... 3answers 2k views ### All polynomials are the sum of three others, each of which has only real roots It was asked at the Bulletin of the American Mathematical Society Volume 64, Number 2, 1958, as a Research Problem, if a Hurwitz polynomial with real coefficients (i.e. all of its zeros have negative ... 4answers 2k views ### Distribution of roots of complex polynomials I generated random quadratic and cubic polynomials with coefficients in$\mathbb{C}$uniformly distributed in the unit disk$|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ... 7answers 3k views ### When is a monic integer polynomial the characteristic polynomial of a non-negative integer matrix? Suppose$P(x)$is a monic integer polynomial with roots$r_1, ... r_n$such that$p_k = r_1^k + ... + r_n^k$is a non-negative integer for all positive integers$k$. Is$P(x)$necessarily the ... 4answers 2k views ###$\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$is a convex function on$[0,+\infty)$? Let$f(x)=\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$, where$u\in\mathbb{Z}^+$and$\binom{x}{l}=\frac{x(x-1)\dots(x-l+1)}{l!}$for all$l\in\mathbb{Z}^+$. Then can we prove$f(x)$is a convex ... 2answers 828 views ### Are there irreducible polynomials with all zeros on two concentric circles? This is somewhat similar to this recent question, but extending in a different direction. Let$f(x)$be an irreducible polynomial of degree$n$with integer coefficients. Call such$f$a bicycle ... 2answers 972 views ### Are these two new ways of representing odd zeta values as integrals known? This is inspired by the same beautiful integral expression for$\zeta(3)$as this question, but goes in a slightly different direction. Writing the original integral in the form $$\int_0^1\frac{x(1-x)}... 0answers 512 views ### Mathieu group M_{23} as an algebraic group via additive polynomials An elegant description of the Mathieu group M_{23} is the following: Let C be the multiplicative subgroup of order 23 in the field F=\mathbb F_{2^{11}} with 2^{11} elements. Then M_{23} is ... 4answers 3k views ### How did Ramanujan discover this identity? Let$$\small F_n=(a+b+c)^n+(b+c+d)^n-(c+d+a)^n-(d+a+b)^n+(a-d)^n-(b-c)^n$$and ad=bc, then$$64*F_6*F_{10}=45*F_8^2$$This fascinating identity is due to Ramanujan and can be found in http://www.maa.... 1answer 1k views ### f(x) is irreducible but f(x^n) is reducible Does there exist an irreducible polynomial f(x)\in \mathbb{Z}[x] with degree greater than one such that for each n>1, f(x^n) is reducible (over \mathbb{Z}[x])? 3answers 1k views ### Changing the signs of the coefficients of a polynomial to make all the roots real We are given a polynomial$$P_n(x):=a_nx^n + a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$with real coefficients. Questions.$\boldsymbol{(i)}$How can we determine if there are$\epsilon_1,\ldots,\...
There are certainly non-monic polynomials of degree 4 with all roots on the unit circle, but no roots are roots of unity; $5 - 6 x^2 + 5 x^4$ for example. Now, for a monic polynomial of degree $n$, ...
Given a polynomial equation $x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$, where $n$ is even and all the coefficients $a_i$ are real, what is the best way to determine whether it has a real root or not? I ...