Questions on the functionality operating on polynomials

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0
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0answers
29 views

Expectation of Largest Roots of Bernoulli distributed coefficient Monomials

I am interested in determining the most probable maximum values of the real roots of polynomials of form $P(x)=\sum_{k=0}^n a_{k} x^k$ where the degree $n$ will have a defined value (say 3,4,5...) and ...
11
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6answers
393 views

Create a polynomial of a given degree

In Mathematica, how can I create a polynomial function in given variables of a given degree with unknown coefficents? That is, I am looking for a function ...
0
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1answer
55 views

Expand a complicated polynomial

I am entirely new to Mathematica. I am wondering whether Mathematica can help me to expand expression like the following $$f(x_1,\dots, x_n) = \left(\sum_i x_i\sum_j x_ix_j^2\right)^3\left(\sum_{i,...
-1
votes
1answer
32 views

How to use Collect[…] with matrices? [on hold]

I have a matrix function $T_4(t)$ which is evaluated as follows: $$\left( \begin{array}{cc} \frac{1}{48} \left(t^4-8 t^2+48\right) & \frac{1}{768} t \left(t \left(t \left(t^3-32 t+64\right)+256\...
2
votes
0answers
114 views

Check Zagier theorem about Mahler's measure

I want to check the following theorem by using Mathematica: (from Heights of Polynomials and Entropy in Algebraic Dynamics, page 22) $\textbf{Theorem}.$ Let $\omega$ denote a primitive $6th$ ...
2
votes
2answers
87 views

Crash after long GroebnerBasis calculation

I am running a long computation of a GröbnerBasis and after some hours the kernel crashes. The memory usage increases enormous, and it crashes, when it reaches somewhat 4 GB RES, however it is ...
0
votes
0answers
40 views

expand[] without multiplying binomial coefficients

I'd like to expand an expression of the form (ax+b)^5 without the binomial coefficents being multiplied out. I would like to see Binomial[5,0], Binomial[5,1] , etc... that intermediate step for ...
6
votes
6answers
405 views

Rewrite a real polynomial in real (but only linear and quadratic) factors

According to my calculus book: "Every real polynomial can be factored into a product of real (possibly repeated) linear factors and real (also possibly repeated) quadratic factors having no real zeros....
7
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1answer
162 views

How to extract all the coefficients of a homogeneous polynomial [duplicate]

Suppose p is a homogeneous polynomial in four variables, say p = x^10 y^10 z^5 w^5 + 3 a x^10 w^20; ...
4
votes
6answers
459 views

Easiest way to extract the coefficient of a polynomial

For a term in a polynomial, say 387 a1^4 a2^3 x^3 y^7 z^100 w^364 what is the most efficient way to extract the coefficient of this term, i.e. ...
2
votes
3answers
100 views

expressing a rising factorial as a polynomial

I have an equation in $x$ as follows: $f(x)=\prod\limits_{j=0}^{k}(j+x)=x(1+x)(2+x)\cdots(k+x)$ I want to express this as a polynomial in $x$, i.e., as $a_0x^0+a_1x^1+\cdots+a_kx^k$ I tried doing ...
7
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6answers
3k views

Series expansion in terms of Hermite polynomials

I am trying to expand a polynomial in terms of orthogonal polynomials (in my case, Hermite). Maple has a nice built-in function for this, ChangeBasis. Is there a ...
2
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2answers
76 views

Expanding rational functions with minimal denominator

I'm working with rational functions and I want to be able to put them in a specific form and then get a list of terms in which the numerators are monomials. Take for example ...
5
votes
1answer
263 views

Finding an analytic solution of a cubic equation

I have been trying to solve a cubic equation $$y=zy^3 + z$$ in $y$ – that is, my desired result is a function $y(z)$ satisfying the equation. Now, there are three solutions of this equation and ...
1
vote
1answer
62 views

How do you collect trigonometric functions in a polynomial?

I have an expression that has various forms of Sin and Cos and I want to collect them specifically so that I can make substitutions. As you can see I cannot figure out how to separate i Cos[theta] ...
0
votes
1answer
48 views

Transforming Determinant to Polynomial Expression

I have determinant which equals zero. In determinant, i have x expressions. i want to transform the determinant to polynomial expression so i can solve that with mathematica. Note : Apologize for ...
3
votes
2answers
62 views

Finding Real Roots and Determining Range

I am interested in determining the minimum and maximum values of the real roots of polynomials of form $P(x)=\sum_{k=0}^n a_{k} x^k$ where $n$ will have a defined value (say 3,4,5...) and $a_k$ are ...
0
votes
2answers
76 views

Finding and plotting a parametric solution to a complicated equation (transcendental, log-polynomial)

I am trying (desperately) to find a way to solve a transcendental equation whose solution $x$ depends on non-numerical parameters $a$ and $b$. And then to produce a ...
1
vote
1answer
224 views

Orthogonalizing polynomials with inner product depending on parameters

I need to orthogonalize the polynomials $h_n(x)=x^{2n}(1+x^2)^{-4S}$ with $x\in\textbf{R}$, $2S\in\textbf{N}$ and $n\in\{1,3,5,\ldots, 4S-1\}$ over the inner product $\langle h_n,h_m\rangle=32\pi S\...
0
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3answers
78 views

Issue finding roots in a polynomial [closed]

I am trying to find the roots of the polynomial (-2 + x)^3 (-2 + x^2) (-4 + x^3) (4 + 2 x^2 + x^4) (-8 - 8 x - 2 x^2 + x^3 + x^4) . I am using the command ...
1
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2answers
73 views

Error In polynomial Root Finding

I have a polynomial in y like so: 2.00855*10^20 + 6.89796*10^20 x y + (5.17347*10^20 + 5.92241*10^20 x^2) y^2 - 1.4806*10^21 x y^3 + 7.77316*10^20 y^4 == 0 I ...
6
votes
1answer
564 views

Polynomial GCD over a ring (with composite characteristic)

I'd like to implement the "Franklin-Reiter Related Message Attack" (see section 4.3 of Boneh's paper). As part of the implementation, I require to compute the GCD of two polynomials over $\mathbb{Z}_N[...
9
votes
2answers
278 views

How can I get the solutions of $x^n-x-t=0$ in hypergeometric form?

$x^3-x-t=0$ has three roots that can be expressed in hypergeometric form, for example $$x_1=-1-\frac{t}{2}{_2F_1\left(\frac{1}{3},\frac{2}{3};\frac{3}{2};\frac{27t^2}{4} \right)}+\frac{3t^2}{8} {...
1
vote
1answer
62 views

Finding a maximum of a Bézier function

Suppose I have a Bézier function $f:\mathbb{R}^2\to\mathbb{R}$ with random coefficients: ...
2
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1answer
79 views

Algorithm for determining factorability

Consider the following polynomial : $P[x,y]:=a_{11}+a_{12}y+a_{13}y^2+a_{21}x+a_{22}x y+a_{23}x y^2+a_{31}x^2+a_{32}x^2 y+a_{33}x^2 y^2$ where the $a_{ij}$ are either $1$ or $-1$. Thus there are $...
2
votes
3answers
1k views

Calculating Taylor polynomial of an implicit function given by an equation

I'd like to write a procedure that will take an equation: F(x,y,z) = 0 chosen variable: x a point: ...
0
votes
1answer
59 views

How to force a Range on Fit?

I have 1001 points between {x,-5, 5}. I wanted to fit a polynomial over the data but when i try: Fit[Flatten[data], {0, x, x^2, x^4, x^5, x^6}, x] ... the range ...
0
votes
2answers
72 views

Wrong numerical results from LegendreP

{Cos[Pi/180] // N, LegendreP[46, 0.9998476951563913`], LegendreP[46, Cos[Pi/180]] // N} give ...
3
votes
1answer
227 views

Is there a package to find ALL exact roots of a polynomial, if they exist?

There are polynomials with roots not expressible with radicals but expressible as trigonometric or other functions, for which Solve[] only returns ...
5
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3answers
331 views

Implementation of a complex recurrence relation for polynomials

I would like to implement the recurrence relation for the polynomials $U_n(x)$ appearing in the large order asymptotics of the Bessel functions. The recurrence in question is: $$U_{n+1}(x)=\frac{1}{...
0
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1answer
63 views

Solving for Polynomial roots

This simple Solve gives the roots of a quadratic: Solve[a x^2 + b x + c == 0, x] However, if I factor the polynomial in terms ...
4
votes
1answer
133 views

Expand power of a polynomial

I'm very new to Mathematica, so excuse my innocence. I have the following expression: $$ \left( \sum_{n=0}^r \frac{(-1)^n}{n!} y^n \right)^f $$ I would like Mathematica to expand out the expression ...
1
vote
5answers
138 views

How find constant term of quadratic with square already completed?

Suppose I have a quadratic polynomial in two variables x and y in which the squares with respect to ...
0
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1answer
42 views

Selecting terms on only one variable from a multiple-variable expression

Say I have a polynomial like $x y^2+15x^2 y+x+3y+10$ and I want to obtain, say, only the coefficient in $x$ alone, namely a 1. Using ...
7
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1answer
252 views

GroebnerBasis without specifying variables

All the examples in the Mathematica documentation specify that the syntax for the GroebnerBasis command is ...
0
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0answers
34 views
0
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0answers
34 views

ApartSquareFree function

I have a doubt concerning the ApartSquareFree function in Mathematica. Roughly speaking, it is supposed to compute the partial fraction decomposition of a rational function $h/g$ with the denominators ...
1
vote
4answers
104 views

Symbolic calculation on roots of polynomial

Given a polynomial like $x^3 + a_2 x^2 + a_1 x + a_0$ with roots $r_i$, I would like to symbolically compute the coefficients of a polynomial whose roots are $r_i^3 + r_i + 1$. How can I do this in ...
4
votes
3answers
269 views

Get the coefficient matrix from a quadratic form

Suppose I have a quadratic form of qf = a x^2 + b y^2 + c z^2 + 2 d x y + 2 e x z + 2 f y z How can I easily to get the symmetric matrix A, such that $X^TAX=qf$?...
2
votes
3answers
170 views

Use NMinimize instead of FindFit for constrained search (of coefficients)

(My problem is more complex, but let us formulate it through this example) I am trying to find the best polynomial approximation to the following function ...
3
votes
1answer
94 views

How to efficiently find only the rational roots of a rational complex polynomial?

I need to find all the rational roots of a bunch of high-order polynomials with rational complex coefficients. Is there an efficient way to do this? My best effort on an example polynomial: ...
4
votes
1answer
107 views

Does Solve[] find ALL the exact roots of rational polynomials?

Does Solve[] find ALL the exact roots of rational polynomials? I've done a bunch of tests where I created an expression with some analytic roots, and Solve[] always found them all. But is the ...
1
vote
1answer
115 views

Factoring an arbitary variable in mathematica

Imagine we have a equation like this gf= 1 + (a3 - a1 x)^2 w1 + (b3 - b2 x)^2 w2 how can I reach the following equation ...
0
votes
0answers
29 views

PolynomialExtendedGCD in 2 variables

Consider a field $R$ and the ring $A=R[y]$. Consider two polynomials $g,h\in A[x]$. I want to obtain $d=\gcd(g,h)\in A[x]$ and two polynomials $s,t\in A[x]$ satisfying the Bézout relation: $sg+th=d$. ...
0
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1answer
79 views

Why are CoefficientRules and MonomialList so slow?

Why is CoefficientRules so slow in this example (v10.2 on OS X 10.11.4)? ...
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3answers
136 views

Alternative forms to Table for iterating over replacement rules

I have a multivariate polynomial x. I get coefficients of various monomials using CoefficientRules, which returns a list of ...
6
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5answers
384 views

Convert polynomial to Chebyshev

I want to convert a polynomial in "standard form" to Chebyshev form. Here's one way to do it: ...
2
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0answers
64 views

Crash on use of CoefficientList

The Kernel of my Mathematica 10.4 seems to crash on certain use of the CoefficientList command. The line CoefficientList[x + y^2, {x, y}] Produces the matrix $$ ...
5
votes
2answers
237 views

Orthogonalize polynomials with respect to Gagliardo seminorm?

For a function $f\colon [-1,1]\to\mathbb{R}$, the Gagliardo seminorm of $f$ is defined to be $$ |f| = \int_{-1}^1\int_{-1}^1 \frac{(f(x)-f(y))^2}{(x-y)^2}\, \mathrm{d}x\, \mathrm{d} y. $$ Given $(x,...