Questions on the functionality operating on polynomials

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7
votes
3answers
3k 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 ...
10
votes
2answers
403 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 ...
5
votes
1answer
867 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) = ...
6
votes
2answers
355 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 ...
10
votes
2answers
231 views
2
votes
3answers
274 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 ...
19
votes
5answers
833 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 ...
1
vote
2answers
181 views

Strange integration result

In Mathematica 8, the Integrate command sometimes strangely integrates polynomials yielding unsimplified (and unexpected) fractional results. As an example, the line: ...
6
votes
1answer
323 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 ...
8
votes
4answers
610 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 ...
6
votes
3answers
568 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 ...
8
votes
4answers
6k 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 ...
5
votes
1answer
535 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 ...
1
vote
2answers
341 views

solving for one solution to a system of polynomials

I'm trying to solve a system of equations: ...
4
votes
2answers
819 views

expanding a polynomial and collecting coefficients

I'm trying to expand the following polynomial ...
8
votes
3answers
1k 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] ...
0
votes
3answers
377 views

Creating simple procedure for The Least-Square $m^\text{th}$ Degree Polynomials

I am CS major, taking Computational Numerical Analysis course. Instructor gave us freedom of choice, we were allowed to use anything or any computer language we picked, I picked Mathematica.This is my ...
10
votes
3answers
383 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 ...
2
votes
2answers
256 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 ...
11
votes
3answers
1k 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.
6
votes
2answers
669 views

Why does Expand not work within a function?

I'm writing this fairly simple function: ...
12
votes
2answers
532 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
2answers
542 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
632 views

Find roots of polynomial in field extension $GF(2^n)$?

How can I find roots of polynomial in extension field $GF(2^n)$?
36
votes
6answers
5k 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 ...
23
votes
2answers
762 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: ...
7
votes
1answer
180 views

GroebnerBasis without specifying variables

All the examples in the Mathematica documentation specify that the syntax for the GroebnerBasis command is ...
9
votes
1answer
440 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 ...
11
votes
1answer
565 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: ...
12
votes
7answers
3k 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 ...
12
votes
4answers
835 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 ...
4
votes
2answers
228 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 ...