Questions tagged [polynomials]
Questions on the functionality operating on polynomials
863
questions
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2
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Tropical Operation Conversions: Multiple Operations
I apologize if my syntax is bad. I looked around, but didn't see anything on this in the stack exchange.
I would like to be able to transform a classical real-valued polynomial to a tropical ...
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1
vote
1
answer
88
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While inside a function body does not work as expected [closed]
I'm trying to write a function that return a randomly created rational polynomial function that has a discontinuities points.
I already have a function that generates random rational polynomial (...
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1
vote
1
answer
60
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How to perform the first and second derivatives of Nth polynomials?
Suppose
$P(x,y)=\displaystyle\sum^{N}_{i+j=0}\alpha_{ij}x^{i}y^{i} \equiv \alpha_{00}+\alpha_{10}x+\alpha_{01}y+a_{11}xy+\ldots+a_{0N}y^{N}$
is a multivariable polynomial of $N$th degree. I want to ...
5
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1
answer
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+50
Rational roots of x^5 + a x + b
I'm validating a formula from these article:
https://www.annals-csis.org/Volume_20/drp/pdf/164.pdf
https://people.math.carleton.ca/~williams/papers/pdf/185.pdf
If irreducible rational equation $x^5 +...
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How to speed up Resultant?
I am experimenting with the polynomials Y and T:
i1 = 9; i2 = 4;
Y = Sum[ x^i y[i], {i, 0, i1}];
T = Sum[ x^i t[i], {i, 0, i2}];
Timing[Resultant[Y, T, x]]
Whith ...
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3
votes
2
answers
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How to know the Galois Group of a polynomial is a solvable group?
ResourceFunction["StauduharGaloisGroup"][2 x^5+3 x^4+10 x^3+15 x^2+8 x+12,x]["GaloisGroup"]
...
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What is the fastest way to check whether a cubic equation is solvable in integers?
My code needs to check whether a lot of low-degree equations (usually quadratic and cubic) are solvable in integers. There are many equations, so the speed is crucial. Let us start with quadratic ...
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Why Cancel and PolynomialGCD treate option Extension in a different way?
I have difficulty understanding the following Cancel and PolynomialGCD examples.
Expected:
...
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4
votes
2
answers
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Rewriting algebraic expressions as combinations of other expressions
I would like to write x^5 + 1/x^5 in terms of x^2 + 1/x^2, x^3 + 1/x^3 and ...
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4
answers
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Constructing a specific polynomial Monomial ordering of $f(z,w)$ to convert to latex
Note: Afraid I did not initially adequately specify the precise Latex format of the algebraic expression. Edited to make more clear.
I am having problems transferring polynomials in two variables $f(...
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Finding the roots of a quartic [closed]
I try to find the roots of $f(x)=x^4+(4a-2)x^3+(6a^2-6a)x^2+(4a^3-7a^2)x+ a^4-2a^3
=0$ with $0<a<1$.
How do i get with Mathematica all solutions (general expressions, real and complex)?
With
<...
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2
votes
0
answers
86
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How to solve this equation analytically?
Why can't Mathematica 13 solve this equation in radicals?
Solve[x^5 - 5*x^4 - 10*x^3 - 10*x^2 - 5*x - 1 == 0, x] // ToRadicals
I also tried to use ...
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2
votes
2
answers
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Three equation solving analytically in Mathematica
Im doing coding in iterative schemes but I do not know to solve three equations analytically in Mathematica step by step. For example I given thre equations in the attached figure. I want to solve ...
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1
answer
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Solving solvable quintics in trigonometric/radicals
If apply resultant to solvable quintic $f(x)$ as $Res_x(f(x),y+Rx+Sx^2+Tx^3)$ and associate output with trigonometric five-angle formula $\cos{5\theta} = 5cos{\theta} - 20cos^3{\theta} + 16cos^5{\...
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2
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How to collect specified monomials in polynomial?
I'm not experienced in Wolfram Language, so couldn't you please help me with its syntax? Given the polynomial expression:
...
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8
votes
2
answers
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Efficient algorithm for calculating polynomial that has roots of a certain form
I have looked into this specific question on Math.SE concerning a more "mechanical" approach to finding a polynomial $p \in \mathbb{Q}[x]$ satisfying $p(\sqrt{2}+\sqrt{3}) = 0$. The user MJD ...
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extremum points
I'm trying to find Min and Max points of the polynomial below, but it seems that the expression is very complicated to mathematical, I've tried a numerical function, such as NMinimize, NRoots, Solve, ...
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0
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Can one collect with respect to two groups of variables?
Here is a toy example of what I want: given
...
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0
answers
74
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Mathematica doesn't simplify 1.` x
I'm debugging a program that uses polynomials with numerical coefficients and it turns out that mathematica does not simplify 1.` x[1]^2 x[2]^2 to ...
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0
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Easy upper and lower bounds for curve genus in Mathematica
I am writing a Mathematica code in which I need, at some point, to compute genus of a lot of curves given by polynomial equations in 2 variables in affine coordinates. Unfortunately, this is possible ...
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0
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38
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Why does Solve give different answers when specifying coefficients at different time?
Context
I would like to find a specific - eventually complex - root of a 4th order polynomial as a function of a purely imaginary parameter $i\lambda$ who is continuous in $\lambda$. I am thus looking ...
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1
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1
answer
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Sort polynomial by monomial degree [closed]
there is an equation :
z = a^2 + a b + a a' + a a' a'' + a b'
and i want to rearrange the equation by sorting the terms by number of variable multiplication within.
If my English is not well, let me ...
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2
answers
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Finding the definite integral of a square root of a polynomial function [closed]
I am trying to find the arc length of the function:
f(x)=-0.005632x^7 + 0.08969x^6-0.5346x^5 + 1.364x^4 -0.8671x^3 -2.005x^2 + 3.038x + 0.4182
I am using this formula:
My boundaries are 0 and 5.
...
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3
answers
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What is the most efficient way to turn a metric formula into a metric tensor?
I have a metric formula:
ds=(-dt^2)*(c3+a3*t)^2+(dxC^2*(t0^2+t1^2)^2)/(4*t0^4)+
(dxM^2*(t0^2+t1^2)^2)/(4*t0^4)+(dxY^2*(t0^2+t1^2)^2)/(4*t0^4)
How do I turn this ...
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Question about NSolve
there!
For some purposes I need to be able to write a code in C# that could find all roots of the equations system, consisting of two-variable polynomial expressions, like the folliwing:
...
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1
vote
2
answers
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why ContourPlot3D does not give plot?
I wanted to plot the intersections using ContourPlot3D in order to observe how do solutions look like, however, the following code does not work:
...
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11
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2
answers
626
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2
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1
answer
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Plotting of a polynomial function via two approaches
I am trying to plot a polynomial via two approaches:
Take CoefficientList of the polynomial and then take Internal`FromCoefficientList, and finally replacing the value of the free parameter with a ...
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1
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1
answer
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Homework from studies! - module computation of an approximating function
Polynomial approximation
The aim of the task: to write a program in the form of a module in the Mathematica® machine code
computation of an approximating function F (·) for an approximated ...
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0
answers
73
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How to solve this quintic polynomial with the solutions in its simplest final form?
Consider the following matrix:
...
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1
answer
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Create a general polynomial function [duplicate]
I search for a simple procedure to create a polynomial function with unknown coefficients.
For example if I asked for the most general polynomial of two variables 'x' and 'y', of second degree, it ...
1
vote
1
answer
65
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Plotting parts of a polynomial
Consider the following polynomial
fun = 2 x + 34 x^2 - 5 y + 4 y^3;
when I try fun[[1]] I get ...
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2
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Efficient way of finding polynomial relation
I have a list p of polynomials of k, and a list of "parameters" x,y,z which are ...
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1
answer
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CoefficientRules numericizing parameters
I ran into a problem with CoefficientRules behaving differently with exact and numerical coefficients, where in the numerical case it behaves as if the entire ...
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2
answers
104
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Problem about vanishing determinant of a matrix with NSolve
When I try to solve my question according to @DanielLichtblau's comments, I encountered another issue, with which I have been struggling for a whole night.
Taking a matrix for example,
...
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2
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Reduce the number of equations
I have following 16 equations with 15 variables (XX is a variable, etc). I know there are only 6 independent variables. How can I reduce the number of equations below to get 9 independent equations?
<...
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1
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Exclude terms in replacement rules application [closed]
Suppose I consider a polynomial of type
A=B x y + C x
and define a replacement rule
rrule=(# /. x y -> 1) &;
The result ...
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votes
0
answers
64
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Is it intended that CoefficientRules permutes exponent vectors when given a monomial order?
I noticed an odd behavior in CoefficientRules when you give it a monomial order:
...
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0
answers
26
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N on equation with numbered variables [duplicate]
Say I have a system of polynomial equations in the variables x[1],...,x[n]. Sometimes want to make such equations numerical (as in: the coefficients become floats) ...
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1
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60
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Minimal polynomial of an element in a polynomial quotient ring
Let $p(z)$ be a (not necessarily irreducible) polynomial in $\mathbb{Q}[z]$. Is there a built-in function I can use for determining the minimal polynomial of an element $\overline{q(z)} \in \mathbb{Q}[...
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95
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Algorithms or Mathematica definition for functions ideals of polynomial Rings
This question was well posed four years ago without answer at Operations on ideals of polynomial rings. I asked again in Nov 2021 and posted possible function defininitions. Lichtblau commented ...
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votes
2
answers
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How can I find the only real and then the smallest root of a 4th-order polynomial?
I just want to solve the below polynomial for real roots only, where I have mentioned the conditions on all variables, a,d,m,L.
...
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answer
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Replace coefficients in a polynomial
I want to replace the coefficients of a polynomial by new coefficients.
For example, my polynomial of variable 'r' is in the picture
I want to replace the coefficients with 'a subscripts' similar to
...
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7
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1
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Solving quadratic equations bug in Mathematica 12.3
There seems to be some sort of bug in mathematica, where sometimes it will treat a quadratic equation as a quartic equation. For example:
...
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0
answers
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Why is PolynomialQ[y^2,x] True? [closed]
Why is PolynomialQ[y^2,x] True? The documentation says PolynomialQ[] returns ...
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vote
0
answers
59
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Expand, simplify, and get rid of Abs[] of a multi-variable polynomial [closed]
Given a polynomial that $poly = ((-1 + t_1^2) (-1 + t_2^2) (1 - t_2^2 + t_2^4))/(\sqrt{t_1^2} \cdot(t_2^2)^{3/2})$. I would like to expand and simplify this polynomial into the form $poly = \sum_{i\in ...
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0
answers
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Mathematica package for computing Macdonald polynomials
I want to implement computation of Macdonald polynomials in mathematica. A similar question was raised in another question 5 years ago (Macdonald-Koornwinder polynomials?), but received no clear ...
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0
answers
60
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Extracting variables [closed]
Similar to this perhaps, but I prefer a built-in function.
I have defined the expression L = a*b*c.
Is there a Mathematica function that can identify and extract ...
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2
votes
1
answer
93
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Factoring a polynomial over a number field
Consider a polynomial x^3-x-1 and let $\alpha$, $\beta$, $\gamma$ be three zeros of the polynomial where $\alpha \in \mathbb{R}$. Since $\beta \not \in \mathbb{Q}[\alpha]$ and the degree of the ...
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