Questions on the functionality operating on polynomials
26
votes
6answers
2k views
Finding real roots of negative numbers (for example, $\sqrt[3]{-8}$)
Say I want to quickly calculate $\sqrt[3]{-8}$, to which the most obvious solution is $-2$.
When I input $\sqrt[3]{-8}$ or Power[-8, 3^-1], Mathematica gives the ...
14
votes
5answers
367 views
How do I reassign canonical ordering of symbols?
I have a big polynomial that evaluates to:
$$A^2 e^2 \phi ^- \phi ^++A e \phi ^- \phi ^+ c_{2 w}
g_Z+\frac{1}{2} A e g h W^- \phi ^+ +\ll13\gg,$$
which is supposed to represent some terms in the ...
12
votes
3answers
257 views
How to keep Collect[] result in order?
For example,
Collect[(1 + x + Cos[s] x^2)^3, x]
gives the result
...
11
votes
2answers
370 views
Surprises simplifying simple polynomials
I came across some somewhat surprising behavior of Simplify today, on something very simple. Let's take two cubic polynomials that we know have the same value:
...
11
votes
4answers
407 views
Is there a way to Collect[] for more than one symbol?
Oftentimes you find yourself looking for polynomials in multiple variables. Consider the following expression:
a(x - y)^3 + b(x - y) + c(x - y) + d
as you can ...
11
votes
1answer
289 views
Rearranging a Polynomial
In Mathematica 8.04 on Windows, I want to display a formula in standard textbook format. The formula is the variance of an $N$-security portfolio. For two securities it is:
...
10
votes
3answers
655 views
Computing the genus of an algebraic curve
How-to compute the genus of an algebraic curve in Mathematica ?
In my case the algebraic curve is explicitly defined by a polynomial.
10
votes
2answers
370 views
Factorizing polynomials over fields other than $\mathbb{C}$
I'd like to take a polynomial in $\mathbb{Z}_5[x]$ of the form $ax^2+bx+c$ and factor it into irreducible polynomials.
For example:
Input...
x^2+4
Output...
...
10
votes
1answer
278 views
Funny behaviour when plotting a polynomial of high degree and large coefficients
I am trying to plot a polynomial of degree 29 on the domain [0,1], with fairly large coefficients:
...
10
votes
2answers
188 views
How can I compute the representation matrices of a point group under given basis functions?
Take the $C_{3v}$ point group for example:
...
10
votes
1answer
342 views
Find roots of polynomial in field extension $GF(2^n)$?
How can I find roots of polynomial in extension field $GF(2^n)$?
9
votes
4answers
315 views
How do I find the degree of a multivariable polynomial automatically?
I have a very simple question which appears not to have already been answered on this forum. Is there built-in functionality that returns the degree of a multivariable polynomial? For example if the ...
9
votes
3answers
197 views
Any efficient way to make complete homogeneous symmetric functions in Mathematica?
We do have elementary symmetric functions, SymmetricPolynomial[k, {x_1, ..., x_n}] .
But I didn't find complete homogeneous symmetric functions.
The induction ...
9
votes
1answer
284 views
Gröbner basis on a particular set of equations
This question is very similar in gist to equation solving with GroebnerBasis, but hopefully when I say that I make the system a little larger I mean little. I have ...
9
votes
2answers
88 views
What are Root objects with multiple polynomials?
In Mathematica 9 a new flavor of Root object with multiple polynomials was introduced. For example,
...
8
votes
4answers
299 views
“Evaluating” polynomials of functions (Symbols)
I want to implement the following type evaluation symbolically
$$(f^2g + fg + g)(x) \to f(x)^2 g(x) + f(x) g(x) + g(x)$$
In general, on left hand side there is a polynomial in an arbitrary number of ...
7
votes
6answers
761 views
How do I replace a variable in a polynomial?
How do I substitue z^2->x in the following polynomial z^4+z^2+4?
z^4+z^2+4 /. z^2->x
...
7
votes
3answers
437 views
Checking if the roots of a function are real
I'm trying to determine if the roots of a function are real. How would you do that?
(In particular I'm interested in verifying that the roots of LegendreP[6, x] ...
7
votes
2answers
265 views
How to deduce a generator formula for a polynomial sequence?
Consider a polynomial sequence $\{p_n\}$ generated by some (simple) rule:
$$
\begin{array}{l}
p_1(x)=x \\
p_2(x)=2 x-x^2 \\
p_3(x)= x^3-3 x^2+3 x \\
p_4(x)=-x^4+4 x^3-6 x^2+4 x \\
p_5(x)= x^5-5 ...
7
votes
1answer
146 views
GroebnerBasis without specifying variables
All the examples in the Mathematica documentation specify that the syntax for the GroebnerBasis command is
...
6
votes
2answers
394 views
6
votes
2answers
242 views
3D Plot: Number of Roots in x of a polynomial in x, a, b and c
I have a polynomial in four variables x,a,b and c. The number of roots of the polynomial in x depends of the choice of a, b and c. I would like to have a 3D-Plot with a, b and c on the axes, while the ...
6
votes
4answers
133 views
How to collect terms with positive powers in polynomial
I am trying to collect all terms with non-negative powers of $x$ in polynomials like
$\frac{1}{x^2}\left(a x^2+x^{\pi }+x+z\right)^2$
First expand the polynomial
...
6
votes
4answers
2k views
Factoring polynomials to factors involving complex coefficients
I've run into some problems using Factor on polynomials with complex coefficient factors. Reading the documentation it looks like it only factors over the ...
6
votes
1answer
341 views
Finding the characteristic polynomial of a matrix modulus n
Given a square matrix, is it possible to calculate its characteristic polynomial modulo n?
Unfortunately, this function ...
6
votes
0answers
57 views
Apart may use Padé method: what's that?
How does Apart work? The page tutorial/SomeNotesOnInternalImplementation#7441 says, "Apart ...
5
votes
3answers
460 views
What function can I use to evaluate $(x+y)^2$ to $x^2 + 2xy + y^2$?
What function can I use to evaluate $(x+y)^2$ to $x^2 + 2xy + y^2$?
I want to evaluate It and I've tried to use the most obvious way: simply typing and evaluating $(x+y)^2$, But it gives me only ...
5
votes
5answers
284 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 ...
5
votes
4answers
511 views
How to get exact roots of this polynomial?
The equation $$ 64x^7 -112x^5 -8x^4 +56x^3 +8x^2 -7x - 1 = 0 $$
has seven solutions $x = 1$, $x = -\dfrac{1}{2}$ and $x = \cos \dfrac{2n\pi}{11}$, where $n$ runs from $1$ to $5$. With ...
5
votes
1answer
131 views
How to express the original ideal elements in the Groebner basis?
Suppose I call
GroebnerBasis[{f1, f2, ...}, {x1,x2, ...}]
The output is a list
{g1,g2,...}
For each $g_j$, there should be ...
4
votes
3answers
375 views
First positive root
Simple question but problem with NSolve.
I need help how to extract first positive root? For example
eq=-70.5 + 450.33 x^2 - 25 x^4;
NSolve[eq== 0, x]
If I ...
4
votes
2answers
278 views
expanding a polynomial and collecting coefficients
I'm trying to expand the following polynomial
...
4
votes
2answers
190 views
How can I make the output from Solve look nice?
I have a problem with presenting solutions. Roots of 4th order polynomials are big expressions. Is there a way to present the roots, s2 and s3, in normal form with some substitutions? Maybe a way to ...
4
votes
1answer
324 views
Polynomial Approximation from Chebyshev coefficients
I would like to expand a function $f(r)$ in the domain $[0,R]$, around the points $r =0$, and $r = R$ in the following manner
$f(r = 0) = \Sigma_{i=0,i = even}^{imax} f_i (r/R)^i$
and
$f(r = R) = ...
4
votes
1answer
89 views
How can I prevent a polynomial from being simplified?
I'm having a problem with polynomials. Let's say I have a polynomial "2x^2 - 5x + 6 - 3x^2" .. How can I check that this expression is not simplified ? Additionally, I would like to locate the ...
4
votes
0answers
90 views
Computing Ehrhart's polynomial for a convex polytope
Is there a Mathematica implementation for computing the Ehrhart polynomial of a convex polytope which is specified either by its vertices or by a set of inequalities?
I am interested in knowing this ...
4
votes
1answer
175 views
How do I get a two-term polynomial with a leading negative sign to display in the correct (i.e. textbook) order?
The first three expressions evaluate as expected and the polynomial is displayed in what I would call "textbook" form. The last expression, however, switches the order of terms. Mathematica employs ...
3
votes
7answers
215 views
Defining a function that completes the square given a quadratic polynomial expression
How can I write a function that would complete the square in a quadratic polynomial expression such that, for example,
CompleteTheSquare[5 x^2 + 27 x - 5, x]
...
3
votes
3answers
211 views
Is it possible to use Composition for polynomial composition?
I want to do this:
$P = (x^3+x)$
$Q = (x^2+1)$
$P \circ Q = P \circ (x^2+1) = (x^2+1)^3+(x^2+1) = x^6+3x^4+4x^2+2$
I used Composition for testing if that could ...
3
votes
2answers
76 views
How to define a polynomial/function from an array of coefficients?
I have the coefficients of my desired polynomial in an array CoefArr (I'm new to mathematica, so I think of everything as arrays, it is actually a list I believe) starting with the constant at index ...
3
votes
1answer
157 views
Small Issue with Chebyshev Derivative Appoximation
I am trying to get approximate the derivative of a function from its Chebyshev expansion.
I start out with the following random function
...
3
votes
2answers
109 views
Better use of Mathematica's PolynomialReduce[]?
I've been using
scPhiDecomp[expr_]:= PolynomialReduce[expr, {x^2-y^2,2 x y}, {x,y}]
which works great on
...
3
votes
3answers
977 views
Get polynomial interpolation formula
I'm attempting to get a polynomial interpolation formula out of Mathematica but I am absolutely lost. I stared out using ...
3
votes
1answer
155 views
How to do the polynomial stuff over finite fields extensions fast?
This question is raised from the problem of package FiniteFields being very slow (please, see the corresponding question): I have had an evidence that Mathematica ...
2
votes
2answers
82 views
Evaluating Polynomials at Grid Points
I am continuing my quest on B-splines. The code below builds a 5x5 matrix out of B-splines, using the BSplineBasis[] routine.
I now want to evaluate the polynomials that are stored in each matrix ...
2
votes
3answers
142 views
2
votes
2answers
165 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: ...
2
votes
1answer
117 views
Implementation of Decompose
I'm curious as to how Decompose works so I decided to use Trace with the option ...
2
votes
2answers
160 views
How to find solutions that yield of root of unity?
I have a polynomial with coefficients that are integer polynomials in another (complex) variable. For example:
1 + (1 - v^2) #1 + (-3 - v^2) #1^2 + #1^3 &
I ...
2
votes
1answer
92 views
Is there any way to force Mathematica to collect a symbol in a polynomial?
Let's say that I have a polynomial like this:
a + b + c
Is there any way that I can get Mathematica to transform it to:
...
