Skip to main content

Partial factorization of multivariate polynomials in terms of given polynomials


I have calculated several homogeneous polynomials in 4,5 or 6 variables $t_1,\dots,t_6$.
I would like to rewrite them as a sum of products of specific lower degree polynomials, which have a meaning in the problem I'm solving.


For example, the simplest polynomial I have is $$ P_4(t_1,t_2,t_3,t_4) = t_1t_2+t_1(t_3+t_4)+t_2(t_3+t_4) $$ which I would like to express in terms of $t_1+t_2, t_2+t_3+t_4, t_1+t_3+t_4$ (only them, although not necessarily all of them). For example, it's easy to see that $$P_4(t_1,t_2,t_3,t_4) = (t_1+t_2)(t_2+t_3+t_4) - t_2^2$$ but I'd prefer something like $$P_4(t_1,t_2,t_3,t_4) = a(t_1+t_2)(t_2+t_3+t_4) + b(t_1+t_2)(t_1+t_3+t_4) + c(t_1+t_3+t_4)(t_2+t_3+t_4)$$


Question: Is there a way to ask Mathematica this sort of partial factorization?


Thank you in advance!


Note: I don't know if a solution exists for my exact polynomials (maybe I have to search for a different interpretation), but I think the general problem is interesting in its own right.




Answer



This is a variant on the suggestion by @Artes. We start by defining a set of substitutions, using some new variables. Create a Groebner basis. Not to worry if that's not a familiar concept, suffice it to say that we use it for purposes of algebraically rewriting polynomials. The thing to know is that the old variables are to be given greater priority than the new, so that they will be rewritten in terms of the "lower" new variables. This is done simply by listing them first, because by default GroebnerBasis will treat them lexicographically by list position. (I'm skimping on what this all means but the code shows what one should actually do.)


subs = {y1 -> (t1 + t2), y2 -> (t2 + t3 + t4), y3 -> (t1 + t3 + t4)};
subspolys = subs /. Rule -> Subtract;
vars = {t1, t2, t3, t4, y1, y2, y3};
gb = GroebnerBasis[subspolys, vars];

Here is your example.


p4 = t1*t2 + t1*(t3 + t4) + t2*(t3 + t4);


We rewrite it as below, using the substitution rules to replace {y1,y2,y3} by the original t variables.


PolynomialReduce[p4, gb, vars][[2]] /. subs

(* Out[17]= -(1/4) (t1 + t2)^2 + 1/2 (t1 + t2) (t1 + t3 + t4) - 
 1/4 (t1 + t3 + t4)^2 + 1/2 (t1 + t2) (t2 + t3 + t4) + 
 1/2 (t1 + t3 + t4) (t2 + t3 + t4) - 1/4 (t2 + t3 + t4)^2 *)

Notice that this is not quite in the form you wrote. One issue is that there is not necessarily a unique way to rewrite p4 in terms of the "new" variables. Moreover it is quite possible that no term order will give a rewrite that corresponds to the desired one even if such a rewrite exists. But this should at least give an idea of how to get close to the desired form. (I note moreover that, per B.GS, one cannot obtain exactly the form you showed.)


Comments

Post a Comment

Popular posts from this blog

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]
Post a Comment
Read more

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]
Post a Comment
Read more

plotting - Mathematica: 3D plot based on combined 2D graphs

I have several sigmoidal fits to 3 different datasets, with mean fit predictions plus the 95% confidence limits (not symmetrical around the mean) and the actual data. I would now like to show these different 2D plots projected in 3D as in but then using proper perspective. In the link here they give some solutions to combine the plots using isometric perspective, but I would like to use proper 3 point perspective. Any thoughts? Also any way to show the mean points per time point for each series plus or minus the standard error on the mean would be cool too, either using points+vertical bars, or using spheres plus tubes. Below are some test data and the fit function I am using. Note that I am working on a logit(proportion) scale and that the final vertical scale is Log10(percentage). (* some test data *) data = Table[Null, {i, 4}]; data[[1]] = {{1, -5.8}, {2, -5.4}, {3, -0.8}, {4, -0.2}, {5, 4.6}, {1, -6.4}, {2, -5.6}, {3, -0.7}, {4, 0.04}, {5, 1.0}, {1, -6.8}, {2, -4.7}, {3, -1.
Post a Comment
Read more