Questions on the functionality operating on polynomials

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31
votes
6answers
4k 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 ...
18
votes
5answers
581 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 ...
14
votes
2answers
199 views

What are Root objects with multiple polynomials?

In Mathematica 9 a new flavor of Root object with multiple polynomials was introduced. For example, ...
13
votes
3answers
482 views

How to keep Collect[] result in order?

For example, Collect[(1 + x + Cos[s] x^2)^3, x] gives the result ...
12
votes
2answers
469 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: ...
12
votes
1answer
468 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: ...
11
votes
3answers
927 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.
11
votes
4answers
269 views

How to replace every possible $A+B$ and $AB$ in expansion of $(A+B)^{10}-A^{10}-B^{10}$ with $x$ and $y$, respectively?

I would like to replace every possible $A+B$ and $AB$ in expansion of $(A+B)^{10}-A^{10}-B^{10}$ with $x$ and $y$, respectively. How to do it with the simplest code in Mathematica? For example, ...
11
votes
2answers
469 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... ...
11
votes
4answers
655 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 ...
11
votes
4answers
609 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
154 views

ToNumberField won't recognize Root as an explicit algebraic number

In Mathematica 9.0.1, it appears that ToNumberField will not always recognize a Root object as an explicit algebraic number. ...
11
votes
1answer
456 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
7answers
2k 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 ...
10
votes
2answers
217 views
10
votes
1answer
532 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
3answers
310 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
388 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 ...
8
votes
4answers
464 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 ...
8
votes
3answers
832 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] ...
8
votes
4answers
4k 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 ...
8
votes
3answers
404 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 ...
7
votes
2answers
350 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
167 views

GroebnerBasis without specifying variables

All the examples in the Mathematica documentation specify that the syntax for the GroebnerBasis command is ...
7
votes
1answer
340 views

Writing an expression as sum of squares of expressions

Suppose we have a symmetric homogeneous polynomial expression $P$ in $X=(x_1,\cdots, x_n)$. I want to check whether there are functions $g(X)$ so that $P$ is of the form $\sum _{1\le i<j\le n} ...
7
votes
0answers
95 views

Apart may use Padé method: what's that?

How does Apart work? The page tutorial/SomeNotesOnInternalImplementation#7441 says, "Apart ...
6
votes
3answers
528 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 ...
6
votes
3answers
1k views

First positive root

Simple question but problem with NSolve. I need help how to extract first positive root? For example: ...
6
votes
4answers
161 views

Lowering the degree of an polynomial with an assumption that the polynomial has a factor x^2+ax+b

Let's assume $ x^{10}+x^5+1 $ has a factor $x^2+ax+b$ Then, If $x^2=-ax-b$, $ x^{10}+x^5+1 =0$. If we successively apply $x^2=-ax-b$ to $ x^{10}+x^5+1 $, we can make $ x^{10}+x^5+1 $ to degree 1, ...
6
votes
2answers
313 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
2answers
589 views

Why does Expand not work within a function?

I'm writing this fairly simple function: ...
6
votes
4answers
335 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
2answers
209 views

Why does PolynomialQ[x^n, x] return False?

From what I can see PolynomialQ will return False whenever some exponent is another variable such as here: ...
6
votes
2answers
134 views

Finding common roots of a specific system of polynomials

Some of my colleges are interested in finding common zeros of the following four polynomials: ...
6
votes
1answer
251 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 ...
6
votes
1answer
142 views

The third argument of the Root function

Consider the roots of an arbitrary indecomposable polynomial: sol = Solve[x^5 + 2 x + 1 == 0, x] The returned expression is ...
6
votes
1answer
209 views

A problem with polynomial root finding

I use the expression ...
5
votes
5answers
822 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
7answers
662 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] ...
5
votes
2answers
186 views

How to find monotonically increasing intervals of a function

I tried this code, but not working Clear[f]; f[x_] := x^3 - 3 x + 2; ForAll[{x1, x2}, x1 < x2, f[x1] < f[x2]] Reduce[%, {x1, x2}, Reals] I expected the ...
5
votes
4answers
1k 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
437 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 ...
5
votes
1answer
141 views

How to substitute the following conditions into an expression?

I have an expression: $p=a\;b\; x + b^2\; y + a\;c\; z$. I want to substitute $a\;b=1$, $b^2 = 2$ and $a\;c = 4$ to obtain $p = x + 2y + 4z$. How can I tell Mathematica to do that? I dont know how to ...
5
votes
2answers
231 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 ...
5
votes
3answers
2k 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 ...
4
votes
2answers
136 views

Finding the parameter that makes real part of an eigenvalue of a matrix 0

I have the following 3x3 matrix: $\left( \begin{array}{ccc} 0.04 -0.4 b & 0 & 0.04 -0.4 b \\ 0 & -0.08-1.2 b & -0.06-0.9 b \\ 1.04 -0.4 b & 2.08 -0.8 b & 0 \end{array} ...
4
votes
1answer
154 views

Factoring a two variable polynomial in a special way

Let $$f=x^9-x^6+4x^5y+2x^3y^2-y^4$$ I would like to factorize $f$ into form: $$(y-F_1(x))\cdots(y-F_k(x))$$ over complex numbers. How can I do it with Mathematica?
4
votes
2answers
144 views

HornerForm of polynomials in terms of E^(i x)

I want to know how to get the HornerForm of the following expression in terms of E^(I x): ...
4
votes
2answers
580 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 ...
4
votes
2answers
606 views

expanding a polynomial and collecting coefficients

I'm trying to expand the following polynomial ...