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# Lazily solving equations in powerseries rings over finite fields

Hello,

I am currently trying to compute coefficients in implicitly defined powerserieses over finite fields. The first is a linear equation of the form a*x+b=0, with a, b given as multivariate lazy powerserieses over the field with 4 elements, and from theoretical considerations I know that there is a solution x=1+... . Even though just writing x=b/a yields an exception, I at least have a more or less efficient way of computing x "by hand", but i would like to be able to define it as a lazy series.

The second application is a bit harder: I want to solve for a powerseries l in one variable over the field with 4 elements that fullfills a system of nonlinear equations, and from theoretical considerations I know that there is a unique solution. My current method is defining l as a powerseries over the infinite polynomial ring over the filed with 4 elements, with coefficients the infinitely many polynomial generators, plugging that into the equations, and solving degree be degree, which takes extremly long...

Does anyone have a suggestion on how to implement this kind of solving as a lazy series?

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## Comments

The question is too abstract. Please provide an actual example.

( 2024-03-19 12:12:48 +0200 )edit

## 1 Answer

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The first case works for me

sage: L.<w, x, y, z> = LazyPowerSeriesRing(QQ)
....: f = exp(w + x + y + z)
....: g = cos(-w + 2 * x - z)
sage: f
1 + (w+x+y+z) + (1/2*w^2+w*x+1/2*x^2+w*y+x*y+1/2*y^2+w*z+x*z+y*z+1/2*z^2) + (1/6*w^3+1/2*w^2*x+1/2*w*x^2+1/6*x^3+1/2*w^2*y+w*x*y+1/2*x^2*y+1/2*w*y^2+1/2*x*y^2+1/6*y^3+1/2*w^2*z+w*x*z+1/2*x^2*z+w*y*z+x*y*z+1/2*y^2*z+1/2*w*z^2+1/2*x*z^2+1/2*y*z^2+1/6*z^3) + ... + O(w,x,y,z)^7
sage: g
1 + ... + O(w,x,y,z)^7
sage: f/g
1 + (w+x+y+z) + (w^2-w*x+5/2*x^2+w*y+x*y+1/2*y^2+2*w*z-x*z+y*z+z^2) + (2/3*w^3-w^2*x+1/2*w*x^2+13/6*x^3+w^2*y-w*x*y+5/2*x^2*y+1/2*w*y^2+1/2*x*y^2+1/6*y^3+2*w^2*z-2*w*x*z+1/2*x^2*z+2*w*y*z-x*y*z+1/2*y^2*z+2*w*z^2-x*z^2+y*z^2+2/3*z^3) + ... + O(w,x,y,z)^7


For the second case, if your equations are polynomials, you can use resultants to find polynomial equations that each variable satisfies then solve from there. If not, I am afraid there's not much better ways... maybe still use resultant symbolically first before using power series ring.

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Asked: 2024-03-19 10:02:04 +0200

Seen: 199 times

Last updated: Apr 03