Mathematics

Questions tagged [algebraic-independence]

30 questions
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Algebraic independence of even polynomials

Fix an integer $k\in\mathbb{N},k>2$ . Are the two polynomials $x^{2k}+y^{2k}+z^{2k}$ and $x^{2}+y^{2}+z^{2}$ algebraically independent? More specifically, could I write the first as a polynomial function of the second? The context of my question is…
Ralph Sabbagh
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Over any field, if some polynomials are algebraically dependent, are their derivatives linearly dependent?

Denote by $P_n$ the space of all polynomials in $n$ variables, with coefficients in a field $\mathbb F$ . A collection of polynomials $(f_1,\cdots,f_m)=\vec f\in P_n\!^m$ is called algebraically dependent if there is some polynomial $g\neq0$ in…
mr_e_man
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Are $\pi$ and $\tan^{-1}\left(2\right)$ rational multiples of each other?

For a proof of quantum universality, I need to show that $\tan^{-1}\left(2\right)$ is not a rational multiple of $\pi$ . How do I show this? I feel like showing algebraic independence over the rationals is hard in general, but is it possible for…
416E64726577
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Can an absolutely convergent infinite linear combination of an algebraically independent family be algebraic?

In the usual definition, a family $\{x_i\}_{i\in I}$ of real (or complex) numbers is algebraically independent if every non-zero polynomial relation involving only finitely many variables is impossible. This “finite character” is essential in the…
KELVIN LAM
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4
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Is $2^{\sqrt2}\cdot\pi$ known to be transcendental number?

Specific Question I know that $2^{\sqrt2}$ and $\pi$ are each transcendental. But is it known that their product or sum are also transcendental? Exposition Schanuel's conjecture is a strengthening of the Lindemann-Weierstrass-theorem . It states If…
Md Asfaque
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4 answers

Algebraic Independence of symmetric power sum polynomials

Let $P_k(X_1,...,X_n) = X_1^k+...+X_n^k$ . My Question is how to proof that the Polynomials $(P_1,...,P_n)$ are algebraically independent. My first try was to imitate the proof of the algebraic independence in of elementary symmetric functions given…
user1072285
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Proof check: Existence of algebraically independent real numbers (over the rationals).

I am not too familiar with infinites and I don't trust myself with them, I am not sure if any step I took supposed something that I shouldn't have. Below I am presenting an exercise in the book "Mathematics++" and my proof. The statement We recall…
Yuumita
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3
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Linear independence vs probabilistic independence

I am trying to solve the following exercise yet I do not know from which angle to attack it. Therefore, I need a hint or two to get me started on both "if" and "only if" implications. Let $V$ be a finite-dimensional vector space over a finite field…
qarabala
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A question on algebraic independence

I just learned about the concept of an algebraic independent set. Here is the definition I started working with. Let $E/K$ be a field extension. A subset $M \subset E$ is said to be algebraically independent over $K$ if every element $\alpha \in M$…
Richard Kruel
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2
votes
1 answer

Preservation of linear independence under field extension

Let $L/K$ be a field extension and $x \in L$ be transcendental over $K$ . Let $A, B$ be $K$ -matrices of same size. I want to show that if $A$ has linearly independent columns over $K$ , then $Ax+B$ has linearly independent columns over $L$ . Here's…
Bubaya
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$(a,b)$ is algebraically independent over $K$ iff $a$ is algebraically independent over $K$ and $b$ is algebraically independent over $K(a)$

I’m using the following definition of algebraic independence: If $K \subset L$ is a field extension, and $(\alpha_i)_{i\in I}$ is a subset of elements of $L$ , we say that the subset $(\alpha_i)_{i\in I}$ is algebraically independent if the $K$…
dahemar
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2
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1 answer

Bilinear map and linearly independent set

let $f$ : ${M}\times{M}$ $\mapsto$ $F$ be bilinear map if $T$ = { $u_{1}$ , $u_{2}$ , .... , $u_{n}$ } is a subset of ${M}$ with $f(u_{i} , u_{j}$ ) = $0$ if i $\neq$ j and otherwise $f(u_{i} , u_{j}$ ) $\neq$ $0$ show that $T$ is linearly…
Liyla morad
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Vector space spanned by incidence vectors of maximal cliques in a graph

I have a undirected graph $(V,E)$ , $V=\{1, \dots, d\}$ , with set of maximal cliques $\mathcal C$ . I am interested in the subspace of $\mathbb R^d$ spanned by the clique incidence vectors $x_C \in \{0,1\}^d$ , $C \in \mathcal C$ where a 1 in place…
Moritz Schauer
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Suppose $\alpha$ is a root of $8x^{4}+4x+3$, then find the integers $n\in \mathbb{Z}$ such that $n \alpha$ is an algebraic integer.

Suppose $\alpha$ is a root of $8x^{4}+4x+3$ , then find the integers $n\in \mathbb{Z}$ such that $n \alpha$ is an algebraic integer. (1) An algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a…
Rotman1729
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Do Egyptian representations follow a matroid structure?

Let $V=(v_1,\cdots,v_n), v_i \in \mathbb N$ be independent with respect to $\frac p q$ if there doesn't exist an indicator vector $\beta = (b_1, \cdots, b_n), b_i \in \{0,1\}$ with $\sum \frac{b_i}{v_i} = \frac p q$ . Clearly if $V \subset W$ and…
Snared
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