All questions with tag [math: polynomials]


Is the ideal $(X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ prime?

Consider the excellent $(f = X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ in the polynomial ring $k[X_0,\ldots, X_n]$. Is this a prime perfect? If so, what is its elevation? I'm stuck attempting to show that $f$ is irreducible.
2022-07-25 20:47:17

Solutions to $z^3 - (b+6) z^2 + 8 b^2 z - 7+b^2 = 0, b\in \mathbb R, z \in \mathbb C$

$z_1 = 1+i$ is a given solution. I guess what I have to find is $z_2$ and $z_3$ in $(z - (1 + i))(z - z_2)(z-z_3) = z^3 - (b+6) z^2 + 8 b^2 z - 7+b^2$. I tried to divide the polynomial by $(z - (1 + i))$, but that didn’t seem to work because of the $b$. According to the Complex conjugate root theorem $z_2 = \overline{z_1} = 1 - i$ is a solution ...
2022-07-25 20:46:44

overdertermined system of polynomial equations

From straight algebra i recognized that if the system of straight formulas are independent of each various other and also if the variety of formulas is greater than the variety of variable, after that the system is irregular (no remedy). Can one expand this to a system of uniform polynomial formulas and also claim that: If the polynomial formu...
2022-07-25 20:19:15

Generalizing an approach to proving AMGM

This trouble is Exercise 5.5.30 of "The Art and also Craft of Problem Solving" by Paul Zeitz. The trouble asks to make use of the identification $$ a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc) $$ to confirm the AMGM inequality. I have actually done this by keeping in mind that the appropriate side amounts to $$(a+b+c)\frac{1}{2}((...
2022-07-25 20:18:46

Generating set for sum of two ideals

Suppose there are two ideals $I,J \in \mathbb{C}[x_1,\dots,x_k]$ and two sets of generating polynomials $\langle f_1, \dots, f_s\rangle$, $\langle g_1, \dots, g_t\rangle$. Now I want to describe $I + J$ with a set of generating polynomials. Is $$\langle f_1, \dots, f_s, g_1, \dots, g_t\rangle$$ a valid generating system?
2022-07-25 20:17:50

How to multiply two polynomials represented by values at distinct points?

I have two polynomials of degree $d$. However, I do not have equations for them. I simply have $d + 1$ distinct points on each polynomial. How would I find the product of these polynomials without deriving an equation for them/finding their coefficients? I would want the product of these polynomials to be represented through $2d+1$ distinct poin...
2022-07-25 20:17:03

Coefficient signs in the sum of successive powers of a polynomial

I'm searching for some structure in the sign variation of the coefficients of: $$P = \sum_{i>0} p^i\enspace,$$ for some polynomial $p \in \mathbb{Z}\langle x\rangle$ with no constant term. I'm interested in whether there is a simple algorithm which, given $n$, will tell me the sign of the coefficient of $x^n$ in $P$. In other words, I'm...
2022-07-25 20:13:59

Degrees of polynomials?

Let $A\left(x\right)$ represent a polynomial with a degree of $n-1$. Split $A\left(x\right)$ into odd and even powers. For example: $A\left(x\right) = 3 + 4x+6x^2+2x^3+x^4+10x^5$ $= \left(3+6x^2 + x^4\right)+x\left(4+ 2x^2 + 10x^4\right)$ More generally: $A\left(x\right) = A_e\left(x^2\right) + \left(x\right)A_o\left(x^2\right)$ where $A_e\left(...
2022-07-25 20:08:24

Polynomial inequality

I found the following problem on a website and would be curious to find a solution. Let $a_1\ge a_2\ge\cdots\ge a_n$ be real numbers such that for all integer $k>0$: $$a_1^k+a_2^k+\cdots+a_n^k\ge 0$$ Let $p=\max\{|a_1|,\ldots,|a_n|\}$. Show that $p=a_1$, and that $$(x-a_1)(x-a_2)\cdots(x-a_n)\le x^n-a_1^n$$ for all $x>a_1$. The f...
2022-07-25 20:05:42

Help with Cardano's Formula

I'm attempting to recognize just how to address cubic formulas making use of Cardano is formula. To examine the method, I expand $(x-3)(x+1)(x+2)=x^3-7x-6$. My hope is that the formula will certainly generate the roots $-1,-2,3$. Yet the formula appears to ruin points: I calculate that $\frac{q^2}{4}+\frac{p^3}{27}=\frac{100}{27}$, therefore the...
2022-07-25 19:52:01

$L_2$-norm representation of the function

Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$ (see for reference I am asking yourself if one can get wonderful depiction of $L_2$ - standard of the function $f^{\alpha}(x)$, particularly $$ \int_{-\...
2022-07-25 19:41:28

How is a degree-$d$ polynomial uniquely characterized by its values at $d+1$ distinct points?

A level - $d$ polynomial is distinctly identified by its values at any kind of $d+1$ distinctive factors. Could a person clarify why the declaration over is always real?
2022-07-25 19:39:32

Interval bisection to find a root of f(x)

I'm trying to recognize Interval bisection. I'm offered a straightforward inquiry in my book, and also I can do the procedure conveniently, I simply do not recognize when to stop. The inquiry is "Use Interval bisection to locate the favorable origin of $x^2 - 7 = 0$, proper to one decimal place" (primarily locate the square origin of 7...
2022-07-25 16:30:32

Antiderivative of Polynomials

I actually like just how distinction is presented for polynomials: Let $P(t) \in A[t]$ : $$D_P(t,s) = \frac{P(t) - P(s)}{t-s} \;\; \in A[t,s]$$ and also the by-product of $P$ is $$P'(t) = D_P(t,t).$$ It appears like a definition from calculus and also it does not entail coefficients of a polynomial. Exists a means to present an antiderivati...
2022-07-25 16:29:20

Is the function "signomial"?

Function $f:(0, \infty)\longrightarrow \mathbb{R}$ is called $\textbf{signomial}$, if $$ f(x)=a_0x^{r_0}+a_1x^{r_1}+\ldots+a_kx^{r_k}, $$ where $k \in \mathbb{N}^*:=\{0,1,2, \ldots\}$, and $a_i, r_i \in \mathbb{R}$, $a_i\neq 0$, $r_0<r_1<\ldots<r_k$, and $x$ is a real variable with $x>0$. My question is simple in the...
2022-07-25 16:27:35

Linear interpolation for finding root of $f(x)$

When using linear interpolation, with similar triangles, to find the root of a function you narrow down the interval the root is in. If $f(1) < 0$ and $f(2) > 0$ then the root is in $[1, 2]$ Then you do linear interpolation to find $1.460, f(1.460) < 0$, then the root is in $[1.460, 2]$ Then again linear interpolation is don...
2022-07-25 16:27:24

Eigenvalues of a matrix and its square

Ok so I messed up my last question, I'll rephrase it: Is there a matrix $A$ of real elements, for which this holds true: $A^2$ has more unique eigenvalues than $A$. If not, then what about if the elements of $A$ were complex numbers? I didn't manage to find such a matrix yet, so I tried proving that it's impossible. I know that the eigenvalues c...
2022-07-25 16:20:39

Characteristic polynomials of matrices

Good day! Given a characteristic polynomial $P$ of matrix $A$ I need to show that the characteristic polynomial $O$ of $A^2$ can't have more different real roots than $P$. I know that the characteristic polynomial for both cases can be calculated like this: $P = |A - \lambda I| = 0$ $O = |A^2 - \lambda^2 I| = 0$ But in a general case with $n*n$...
2022-07-25 16:16:47

The coefficient of $x^{18}$ in $(1+x^5+x^7)^{20}$

I was inquired about a straightforward inquiry that is: "What is the coefficient of $x^{18}$ in $(1+x^5+x^7)^{20}$? Usually, we understand that ; $$(x+y+z)^n= \sum_{n_{1}+n_{2}+n_{3}=n}\left(\frac{n!}{n_{1}!n_{2}!n_{3}!}\right)x^{n_{1}} y^{n_{2}}z^{n_{3}} $$ So, this inquiry can be addressed conveniently based upon above formula. May I ask,...
2022-07-25 16:16:25

What does it mean for a subset of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent?

What does it suggest for a part of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent? Specifically I would certainly such as to recognize the solution thereof which worries the bit of a surjective ring isomorphism.
2022-07-25 15:49:59