A question on polynomials
Let $f(x,y) = ({x^2} + {y^2})p{(x,y)^2} – q{(x,y)^2}$ and $p$ and $q$ are two polynomials. Is it true that; $frac{{partial f}}{{partial x}}$ and $frac{{partial f}}{{partial y}}$ don’t have a common...
View ArticleProving irreducibility; What is this method and what is the logic behind it?
The only two methods I know are Eisenstein’s method Irreducibility modulo $n$ Now, I am asked the following question Show whether or not $p(x)=x^5-5x^4+10x^3-7x^2+8x-4$ is irreducible over $mathbb{Q}$...
View ArticleA polynomial $P(x)$ of degree $5$ with lead coefficient one,increases in the...
A polynomial function $P(x)$ of degree $5$ with leading coefficient one,increases in the interval $(-infty,1)$ and $(3,infty)$ and decreases in the interval $(1,3)$. Given that $P(0)=4$ and $P’(2)=0$,...
View ArticleShowing reducibility of a polynomial in a Discrete Valuation Ring
Let $R$ be a complete discrete valuation ring with uniformiser $pi$. I would like to show that a polynomial $f$ in $R[X]$ is reducible. Does it suffice to show that $f$ is reducible in...
View ArticleWhere does randomness help when deciding algebraic geometry over $mathbb{C}$?
If we have a single straight line program expressing a multivariate polynomial equation with integer coefficients, the Schwartz-Zippel lemma gives a simple randomized algorithm for deciding whether the...
View ArticleWhy polynomial $psi^top(t) A^{-1} psi(t)$ attains maximum on $[-1, 1]$ at $pm...
Problem. Let $psi(t) = (1, t, t^2, ldots, t^{p-1})$ – a polynomial basis. Suppose there is a matrix $$ A = int_{-1}^1 psi(t) psi^top(t) dt, text{i.e. } A_{ij} = [2 , | , i+j] cdot dfrac{2}{i+j+1}...
View ArticleHow to find a non-zero point of a non-zero polynomial of low degree?
Given a circuit that computes a polynomial $P(x_1 dots x_n)$ of low formal degree over some large field $mathbb{F}$. Moreover, given a point $X in mathbb{F}^n$, such that $P(X) neq 0$. Can one...
View ArticleHow to show that $X^p-tinmathbb{F}_p(t)[x]$ is irreducible? [duplicate]
This question already has an answer here: Why this polynomial is irreducible? [on hold] 2 answers
View ArticleCounting the number of roots of multivariate polynomials?
The equation of a circle is well known $$(x-x_0)^2+(y-y_0)^2 – r^2 = 0$$ It has a solution all along the circle with midpoint $(x,y) = (x_0,y_0)$. We also know that $ab = 0$ whenever any of $a$ and/or...
View Article$R$ commutative ring with unity , does polynomials with unit leading...
Let $R$ be a commutative ring with unity , consider the polynomial ring $R[x]$ , let $mathcal P_n:={f in R[x] : f=0$ or $deg f le n}$ , so $mathcal P_n$ is a finitely generated module over $R$ . Let $a...
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