# Tagged Questions

Questions about the set of values at which a given function evaluates to zero. For questions about "square roots", "cube roots", and such, consider using the (radicals) and (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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### How to solve a quintic polynomial equation?

I know that not all quintics are solvable. But how do I identify the class of solvable ones?
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### Solutions of an equation of degree $n>4$

I know that the Abell-Ruffini theorem prove that we cannot solve a general equation of degree $n>4$ with radicals. But I've read that quintic equations can be solved by means of elliptic modular ...
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### Does there exist a closed form expression for roots of polynomials of the form $x^n + bx + c$

Where $n \in \mathbb{N}$. $n$ need not be less than five.
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### Companion matrix of bivariate polynomial

A polynomial in one variable can be expressed as a companion matrix, of which the eigenvalues are the roots of the polynomial and which can be found by using e.g. QR decomposition or power iteration. ...
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### Why does Ridders' method work as well as it does?

I've just read section 9.2.1 in Numerical Recipes Ed. 3 (Press et al. 2007), which describes Ridders' method of root finding. I understand that allowing for some curvature of the function by ...
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### What is the number of zeros of antiderivatives of $(x-1)(x-2)^2(x-3)^3(x-4)^4$?

For each $x \in \mathbb{R}$, let $f(x) = (x-1)(x-2)^2(x-3)^3(x-4)^4$. This defines a function $f : \mathbb{R} \to \mathbb{R}$. There is a unique natural number $k$ such that every antiderivative of $f$...
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### Showing that the n first derivatives of (x²-1)^n have at least r roots (for the r-th derivative)?

I have f(x) = (x²-1)^n. I want to show that, for r = 0,1,2,...,n, the r-th derivative is a polynomial (that's easy to show) that has no fewer than r distinct roots in (-1,1). I guess I need to use ...
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### Trigonometric functions of angle fractions

I've just encountered a problem that seems to me interesting enough so that some result exists on the subject. I was working on a problem in complex analysis, in which I needed the fifth root of a ...
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### Roots of the derivative as symmetric (?) functions of the roots of the polynomial

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...
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### Zeros of $\frac{1}{B(xi)^{1/2}}((iA)^{ix})(ix)^{ix}+ \frac{1}{B(-xi)^{1/2}}((-iA)^{-ix})(-ix)^{-ix}=H(x)$

What would be the zeros of the following function? $$\frac{1}{B(xi)^{1/2}}((iA)^{ix})(ix)^{ix}+ \frac{1}{B(-xi)^{1/2}}((-iA)^{-ix})(-ix)^{-ix}=H(x)$$ This function is real and I believe it is equal ...
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### Existence of a Root of Elementary Monomials

Let $m_\lambda(X_1(t),X_2(t),...X_N(t))$ be a monomial symmetric function with partition $\lambda$. For example: $$m_{(3,1,1)}(X_1(t),X_2(t),...X_N(t)) =X_1^3X_2X_3 + X_1X_2^3X_3 + X_1X_2X_3^3$$ ...
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### Existence of n distinct (real) roots of an orthogonal polynomial

I'm trying to get my head around the proof that an orthogonal polynomial ($P_n$ say) has at least n distinct roots. My understanding of the proof http://en.wikipedia.org/wiki/Orthogonal_polynomials#...
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### Monotonic roots

Consider we have a stricktly increasing positive sequence $\lambda_n$ and the following sixth order algebraic equation for every $n\in \mathbb{N}$, $$\zeta s^6-s^4+\lambda_n^2=0,$$ where $\zeta$ is a ...
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### Finding all complex roots of an equation with exponentials.

I know that $$(-1)^x + 2^x - 2 x - 1 = 0$$ has a single real root $(x =3)$ and an infinite number of complex roots whose real part appears often negative. Don't the complex roots also have their ...
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### Solve transcendent modified Bessel function equation

I would like to solve an equation of this type: $$A-BK_0(\kappa r)-CK_1(\kappa r)=0\ \ \ \text{with}\ A,B,C,\kappa\in\mathbb{R},\ \text{known}$$ i.e. I am looking for the zeros of the l.h.s. I am ...