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Questions tagged [polynomials]

This tag is used 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.

0
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2answers
14 views

Does adding a term of a different magnitude that is multiplied by 0 change the degree of the function?

Consider some quadratic function $y=ax^2 + bx + c$. Consider changing the function such that $z = 0x^3 + ax^2 + bx^2 + c$. Can it be said that $y = z$? Can it be said that $z$ is a cubic function, ...
0
votes
0answers
8 views

Finding optimal knots for function approximations

I would like to approximate a continuous (complex) function $f(x)$ in the interval $[a,b]$ $ (x\in\mathbb{R})$ by local polynomial functions of order $3$ (cubic Hermite spline or cubic C2 spline). Is ...
0
votes
1answer
37 views

Number of points needed to determine a sparse univariate polynomial over a large prime field with known support

Consider a sparse polynomial $f \in \mathbb{F}_p[x]$, of maximum degree $k$, with a known support of $t$ terms. That is, we know a set $I \subseteq \mathbb{N}$ such that $\#I = t$ and $f(x) = \sum_{i \...
1
vote
1answer
21 views

How to find shorter encoding of $256$ bit number

Wondering if you could encode a number such as $2^{256}$, as a polynomial equation or some other encoding that would make it shorter than its actual value written out in decimal notation which is ~70 ...
1
vote
0answers
17 views

What is the Computational Complexity of the Elementary Symmetric Polynomials

The elementary symmetric polynomials in $n$ variables, $e_k(X_1,\dots,X_n)$, are defined implicitly by $$(X-X_1)(X-X_2) \cdots (X-X_n)=\sum_{k=0}^n (-1)^k e_k(X_1,\dots,X_n) X^{n-k}, \quad 1 \leq k \...
1
vote
1answer
33 views

Show that $a_{k-1}a_{k+1}\le a_k^2$ where $P(x) = \sum_{k=0}^n a_k X^k$ is a real polynomial with real roots, and $0<k<n$

the question is in the title. I know that this is a direct consequence of Newton's inequalities but I'm looking for a proof without using it. A hint was given to solve it : Show that $$ P'^2 - P'...
-1
votes
1answer
23 views

Problem Regarding Polynomial and Division Algorithm [on hold]

Divide the polynomial $ P(x) = x^4+3x^3-7x^2+11x-1 $ by $ x^2+2 $ and write your result in the form of $ P(x) = (x^2+2)Q(x)+cx+d $. Thanks!
-1
votes
1answer
17 views

Question regarding polynomials and common factors

Let P(x) and Q(x) be distinct polynomials with a common factor (x-a). Show that R(x)=P(x)-Q(x) will have the same common factor.
1
vote
3answers
32 views

Question Regarding Remainder Theorem and Polynomials [on hold]

Show that when the polynomial $f(x)$ is divided by $(x-a)(x-b)$ where $a \neq b$, the remainder is $ \frac{(x-a)f(a)-(x-a)f(b)}{a-b} $. Thanks!
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0answers
30 views

Do these polynomials with harmonic number-related coefficients lie in some particular known class?

I've generated a set of univariate polynomials ($b=1,2,\ldots$) in $v$ of degree $b-1$. The constant term and the coefficient of $v^{b-1}$ is simply $H_b$, the $b$-th harmonic number. The ...
4
votes
1answer
69 views

Proof of separability of polynomials without derivatives

Is there a known proof without differentiating that proves that all irreducible polynomials over $\mathbb{Q}$ are separable? (Or even better, for all fields of characteristic $0$.) EDIT: As people ...
0
votes
2answers
37 views

Product of cubic root differences

Given a cubic equation $x^3 + ax^2 + bx + c$, with three real roots $r_1, r_2, r_3$, how can we express $(r_1 - r_2)(r_2 - r_3)(r_3 - r_1)$ in terms of $a,b,c$ and how do we prove it using Vieta's? ...
1
vote
0answers
12 views

Degree of non-degenerated Boolean function over field of prime order

This problem occurred when we were trying to analyze the degree of a Boolean function over fields of different prime order. Problem: For any prime $p$, assume $n$ is large enough, then any non-...
1
vote
4answers
50 views

Show that the set $\{u_1, u_2, u_3,\dots, u_k\}$ is linearly independent

Let $\{u_1, u_2, u_3,\dots, u_k\}$ be real polynomials: $u_1(x)=1$ $u_2(x)=1+x$ $u_3(x)=1+x+x^2$ $\dots$ $u_k(x)=1+x+\dots+x^{k−1}$ with $k$ a positive integer. Show that the set $\{u_1, u_2, ...
2
votes
1answer
33 views

Write formula in elementary symmetric polynomials

Consider the expression $$ \prod_{i\in I, j\in J} (x-\alpha_i - \beta_j) $$ as polynomial in $x$. As this expression is symmetric in the $\alpha_i$ and in the $\beta_j$, you should be able to write ...
1
vote
2answers
39 views

Does the vector space spanned by a Gröbner basis depend on the monomial order?

Let $k$ be a field, $n$ a natural number, and $I$ an ideal of $k[x_0,\dots,x_{n-1}]$. Given a monomial order $M$ on $k[x_0,\dots,x_{n-1}]$ let $G_M$ be the Gröbner basis of $I$ with respect to $M$ and ...
0
votes
1answer
61 views

Definite integral as dot product

I was reading examples of dot products and I came across the following: In the vector space $P_n(\mathbb{R})$ of polynomials with degree less or equal to $n$ we consider the following dot product:$$...
0
votes
0answers
21 views

Closed-Form Formula For Higher Powers of an Algebraic Number

Let $\mathbb{K}$ be a finite-degree algebraic field extension of $\mathbb{Q}$, and let $\xi$ be algebraic over $\mathbb{K}$ of degree $d$ minimal polynomial: $$X^{d}-\sum_{k=0}^{d-1}\alpha_{k}...
3
votes
0answers
68 views

Question about Lang's Chapter 6 Theorem 9.1

I am an undergraduate working through Chris Hall's result about infinitely many twin irreducible polynomials over finite fields. He begins his argument with a lemma, If $q \equiv 1$ mod $l$ for ...
-1
votes
1answer
34 views

Finding a cubic polynomial that, when divided by $x+1$ and $x-1$ has respective remainders $-1$ and $+1$, and whose value at $x=0$ is $1$. [on hold]

Help I got this question in class... A monic cubic polynomial, when divided by $(x-1)$, has the remainder $-1$, and when divided by $(x+1)$ has the remainder $1$. Find the polynomial in the form $...
-2
votes
1answer
41 views

problem of factorization equation

I think the question is wrong! am I right? $x^2+17y^2-8xy-6y+9 = 0$ $2x-y=?$
1
vote
0answers
15 views

Do these lie in a particular class of interestingly-structured (univariate) polynomials?

I have a series of polynomials ($i=1,2...,14$) of the form \begin{equation} 1-v \end{equation} \begin{equation} 1-v^2 \end{equation} \begin{equation} -v^3-\frac{27 v^2}{11}+\frac{27 v}{11}+1 \end{...
1
vote
1answer
69 views

Closed form of the real zeros of $x^{n+1}-2x^n+1$ for positive integer $n$

I was wondering: What are the real zeros of the function $$x^{n+1}-2x^n+1$$ where $n$ is a positive integer? Obviously, there is a zero at $x=1$. But, if $n$ is even, there are two other ...
0
votes
2answers
73 views

Is this a polynomial? $Y = (256X^8 + 81X^4)^{1/4}$ [on hold]

Is this a polynomial over $\mathbb R$? $$Y = (256X^8 + 81X^4)^{1/4}$$
2
votes
2answers
62 views

How many roots does an exponential polynomial have?

Let $s$ be a complex variable and consider two polynomials with real coefficients: $$A(s) = s^n + a_{n-1}s^{n-1}+\ldots+a_1s+a_0,$$ $$B(s) = s^m + b_{m-1}s^{m-1}+\ldots+b_1s+b_0,$$ where $n \ge m$. ...
2
votes
1answer
45 views

Proving inequality involving radical

I have tried many examples for the following inequality in Mathematica. It is likely true. I need some help proving it. For $x_1, x_2, y_1, y_2 \geq 0$, \begin{align*} 6(x_1^5y_1 + x_1y_1^5) + 4(...
-3
votes
1answer
63 views

Write $x^{n}−1$ as a product of two polynomials [on hold]

Given $n>1$, write the polynomial $x^n−1$ as a product of two polynomials each of degree less than $n$. Edit (as I didn't provide enough info initially): I don't know where to start with this ...
2
votes
0answers
36 views

Why are polynomials of even powers better for Pollard's rho?

Taking all $C(900,2)$ combinations of the first 900 prime numbers, I defined $N = pq$, where $p$ and $q$ are a combination of primes. Then I factored $N$ using Pollard's Rho, counting how many ...
0
votes
2answers
29 views

Find rational representation of a power series

I need to find a rational function $\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials, which value is the same as $\sum_{n = 0}^\infty (2n+1)z^{2n}$ on its convergence domain. I found $\rho=...
2
votes
0answers
21 views

Is there a technique to simplify a system of polynomial equations using known solutions?

If I am working with a polynomial and guess a root $x_0$, I can divide the polynomial by $x-x_0$ to obtain a simpler polynomial without that one root. Can I do something similar for a system of ...
0
votes
0answers
39 views

Factoring a general biquartic into two quartics

Let $W_0$, $W_1$, $W_2$, and $W_3$ be known real numbers. I have to solve a biquartic equation: \begin{equation} z^8+W_3z^6+W_2z^4+W_1z^2+W_0=0 \notag \end{equation} Of course I could solve the ...
0
votes
1answer
13 views

Conditions when the system of symmetric quadratic equations has a solution

suppose we have a system of quadratic equations of the following form \begin{align} \boldsymbol x^T\boldsymbol A_i (\boldsymbol x + \boldsymbol b)=y_i\ ,\quad i=1,\cdots,M \end{align} where $\...
2
votes
1answer
22 views

What does $\mathbb C(z)[x]$ denote?

I am currently reading a German book about complex analysis $($to be precise K. Fritzsche: Grundkurs Funktionentheorie$)$. However, within the $4$th chapter he deals with elliptic functions and ...
3
votes
1answer
100 views
+50

An open cover of $\mathbb{R}^n$ and $\mathbb{C}^n$

Consider the following subset of $\mathbb{R}^n$: \begin{eqnarray}V_i:=\{(p_1, \cdots, p_n)\in\mathbb{R}^n|x^n-p_1x^{n-1}+p_2x^{n-2}-\cdots+(-1)^np_n=0\text{ has at least one root with multiplicity at ...
3
votes
0answers
75 views

A peculiarity of Eisenstein polynomials

I've tested polynomials of degree 6 with random integer coefficients $|a_i|<50$ in test-series of $10,000$. The probability of a random primitive polynomial of the kind to be reducible seems to be ...
-2
votes
2answers
50 views

Can anyone help me with these Polynomial Questions? [on hold]

I am really stuck on this polynomials question and would really appreciate some help. any suggestions for software to use when graphing would also be a great help.
1
vote
0answers
27 views

Number of roots and degree of polynomial [duplicate]

Let $F$ be a field, and $p \in F[X]$ let be a polynomial of degree $n.$ There exists some field extension where $p$ has $n$ roots. Do you know the proof of following statement?
2
votes
1answer
39 views

Kummer ring - special monic polynomial with zero at root of unity

Let $ \zeta_n = e^{2 \pi i / n} $ be the n-th root of unity. Let $$ P(z) = \sum_{k = 0}^{n-1} s_k z^k $$ be a monic polynomial over $ z \in \mathbb{C} $, specified by integer coefficients $ s_k \...
2
votes
2answers
29 views

Roots in a polynomial ring

If we have a polynomial in a polynomial ring, are its "roots" only valid if the roots are in the ring itself? i.e. For $x^2+1 \in \mathbb{C}[x]$ we have roots $i$ and $-i$. But if we consider it as $...
2
votes
3answers
58 views

How to compute remainder of division of $P(x)$ by $x^2 -3x+2$? [duplicate]

The remainder of division of $P(x)$ by $x^2−1$ is $2x+1$, and the remainder of division of the same polynomial by $x^2−4$ is $x+4$. Compute the remainder of division of $P(x)$ by $x^2−3x+2$. I will ...
1
vote
0answers
15 views

Minimal polynomials of primitive elements compared to normal elements

Let $gcd(n,q)= 1$ Consider $x^n - 1 \in \mathbb{F}_{q}[x]$, and let $\mathbb{F}_{q^t}$ be a splitting field for $x^n - 1$ over $\mathbb{F}_q$. Then, $\mathbb{F}_{q^t}$ contains a primitive $n^{th}$ ...
0
votes
1answer
13 views

Show that if $p(x)$ is not divisible over integral domain then $p(x+a)$ is also not divisible. [duplicate]

Let $a\in R$. How to show that if a polynomial $p(x)\in R[X]$ where $R$ is an integral domain is not divisible if and only if $p(x+a)$ is not divisible?
3
votes
1answer
71 views

Is the Pisot Triangle series known?

The Kepler triangle is built with powers of $\sqrt\phi$ to make a right triangle. The supergolden ratio can make a 120° triangle. It turns out that most Pisot numbers (Mathworld, Wilkipedia) 1 to 4 ($\...
4
votes
1answer
96 views

Symmetry in function given by double sum

I had to deal with this function: $$ f_n(x_1,x_2)=(x_2-x_1)^{n-1}\sum_{m=0}^{n-1}\sum_{j=0}^{n-m-1}C(n,m,j)\left(\frac{x_2}{x_2-x_1}\right)^m\left(\frac{x_2(1-x_1)}{x_2-x_1}\right)^j $$ where $$C(n,...
1
vote
1answer
36 views

Factor Theorem for Multivariate Polynomials [duplicate]

I am looking for neat ways of proving the following theorem: Let $F$ be a field and let $f \in F[t_1, ..., t_n]$ be a polynomial. If $f(\pmb{u}) = 0$ for some $\pmb{u} \in F^n$, then $f$ lies in ...
0
votes
1answer
26 views

Polynomial Ring modulo Ideal is Polynomial ring of cosets of indeterminates.

I wonder if for any arbitrary ideal $I \leq K[X_1,\ldots,X_n]$, the following is true: $$ K[X_1,\ldots,X_n] \text{ mod } I = K[X_1 + I, \ldots, X_n +I].$$ If so, how can one show that?
0
votes
0answers
28 views

Equation/formula/algorithm for figuring out binary polynomial equation from integer

Say you have this sequence of examples: 2^0 = 1 2^1 = 2 2^1 + 2^0 = 3 2^2 = 4 2^2 + 2^0 = 5 2^2 + 2^1 + 2^0 = 6 2^3 - 2^0 = 7 2^3 = 8 ... There is some equation ...
0
votes
0answers
24 views

System for producing shortest form of natural numbers

Wondering what the way of producing the shortest length string form of the natural numbers $\mathbb{N}$. I am thinking about this in terms of a binary representation (base 2 representation), but I ...
1
vote
0answers
31 views

Show that there exist $a,b \in K [X_1,X_2,\cdots,X_n]$ and $d \in K[X_1,X_2,\cdots,X_{n-1}]$ such that $aF+bG = d.$

Let $K$ be a field. Let $F,G \in K [X_1,X_2,\cdots,X_n]$ be two polynomials which are relatively prime to each other. Show that there exist polynomials $a,b \in K [X_1,X_2,\cdots,X_n]$ and $0 \neq d \...
0
votes
0answers
21 views

Can a Laurent polynomial be written as one exponentiation?

Polynomial with positive exponents can be written as one exponentiation with a constant term. For example, the quadratic function $f(x) = x^2-10x+35$ can be written as $f(x) = \left(x-5\right)^2+10$. ...