Questions tagged [polynomials]

For both basic and advanced questions on polynomials in any number of variables, including, but not limited to solving for roots, factoring, and checking for irreducibility.

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If a sequence is generated by a $\mathbb{Q}$-polynomial passed mod $p$, can we find an appropriate polynomial over an extension of $\mathbb{F}_{p}$?

If we have a polynomial that takes integer values for integer inputs, we can take its outputs at integer inputs and pass them $\text{mod }p$. However, my understanding is that the coefficients of the ...
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Determining the signature of a Matrix based on the characteristic polynomial.

Let $A$ be a hermitian matrix with the characteristic polynomial $p_A=a_0+a_1x+...+a_nx^n.$ Furthermore let $p$ be the number of sign changes in the sequence $\{a_0,a_1,...a_{n-1},1\}$ and $q$ be the ...
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Compare weak and finest locally convex topology on $\mathbb{R} [x]$

Let $V = \mathbb{R} [x] \cong \bigoplus_{\mathbb{N}} \mathbb{R}$ be the vector space of univariate polynomials, or the space of real sequences that have all but finitely many elements equal to zero. ...
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In what sense is $\Bbb R(x)$ an "instantiation" of the hyperreals?

I'm teaching myself about hyperreal numbers. My main motivation for doing so is that they include infinite numbers, whose existence I hear disputed & doubted often as "quantifying the ...
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$\mathrm{Res}(f, g, h)$ as a product of three resultants?

I'm currently working with resultants, which I define as follows: let $k$ be a field, $V$ be a 2-dimensional vector space over $k$, and let $S^dV^*$ be the $d$-th symmetric power of dual space, i.e. ...
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Generalizing Ramanujan cubic denesting formula to higher powers

We have the following theorems for denesting radicals of degree 2 and 3 : Denesting theorem for degree 2 : If $\alpha, \beta$ are the roots of the equation, $$x^2-ax+b = 0$$...
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Finding Polynomials with negative coefficients

There is a well known mathemagic trick where a volunteer from the audience comes up with a single variable polynomial with positive integer coefficients and no like-terms. The mathemagician then can ...
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Finding a polynomial satisfying $f(a)=A$, $f'(a) = 0$, $f(c)=C$, $f'(c)=0$, $f'(b)=m$

I'm looking for a curve (polynomial?) which satisfies the following constraints: $$f(a) = A, f'(a) = 0$$ $$f(c) = C, f'(c) = 0$$ $$f'(b) = m$$ $$A < C , a < b < c, m > 0$$ ...
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Can $p\big(\frac{z+z^{-1}}{2}, \frac{z-z^{-1}}{2i}\big)$ vanish for every $z$, for some polynomial $p$ in two complex variables?

Is there some non-vanishing polynomial $p$ in two complex variables, such that $p\big(\frac{z+z^{-1}}{2}, \frac{z-z^{-1}}{2i}\big) = 0$ for every non-zero complex number $z$? I'm only interested in ...
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Easier way to solve equation systems of $a+b+c+\cdots{}= 1$, $a^2 + b^2 + c^2+\cdots{}=2$ and so on without having to crunch massive expressions

I study at below college level. I have been trying to solve certain systems of equations involving $n$ equations of $n$ unknowns. For example, for $2$ unknowns, the problem is \begin{align} a^{\...
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Finding functions $f, g, h : \mathbb R_{> 0} \to \mathbb R_{> 0}$

Undergraduates at my university showed me this problem, which I found intriguing and now want to see the solution of: Find all functions $f, g, h : \mathbb R_{> 0} \to \mathbb R_{> 0}$ such ...
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How to find modular roots of $x^{22}-2x^{11}-x+2$ (to show it has more than $22$ solutions by CRT).

Consider a polynomial $P$ defined by $P(x)=x^{22}-2x^{11}-x+2,$ how to show that there exists an integer $n\geq1$ such that the equation $P(x)\equiv0$ modulo has more than $22$ solutions modulo $n?$ *...
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Find the number of positive integers $n,$ $1 \le n \le 100,$ for which $x^{2n} + 1 + (x + 1)^{2n}$ is divisible by $x^2 + x + 1.$ [duplicate]

Find the number of positive integers $n,$ $1 \le n \le 100,$ for which $x^{2n} + 1 + (x + 1)^{2n}$ is divisible by $x^2 + x + 1.$ I cannot figure this one out. If $x^{2n} + 1 + (x + 1)^{2n}$ is ...
1 vote
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$P(x)$ is odd-deg polynomial with real coefficients. Show using Induction that $P(P(x))=0$ has atleast as many distinct real roots as the $P(x)=0$.

PROBLEM Let $P(x)$ be an odd degree polynomial in $x$ with real coefficients. Show using Induction that the equation $P(P(x))=0$ has at least as many distinct real roots as the equation $P(x)=0$. MY ...
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If $f(x)$ is irreducible, is $f(x^k)$ irreducible?

Let $f(x)\in\mathbb{Z}[x]$ be an irreducible polynomial of degree $\ge 2$. Is it true that $f(x^k)$ is irreducible for $k\ge 2$? If not true, under what hypothesis, we can gurantee positive answer? ...
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Are there other expressions similar to polynomials in their ease of calculation and customizability?

I have a bunch of 2D squiggles and I need to get their y value based on an input x value. All of them can be calculated through some kind of y = f(x), as in each y value has one and only one ...
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