kejv2's profile - activity

Anyway, I realize now that this whole exercise (while interesting in itself) is probably pointless since it seems that c

Sorry, I made a copy&paste error probably but my computations were correct, i.e. this to_old( e.from_polynomial( (a^

Sorry, I made a copy&paste error probably but my computations were correct, i.e. this to_old( e.from_polynomial( (a^

One of the possible options I tried was to first reduce by e[1]: e.from_polynomial(f).reduce([a+b+c]).specialization({a:

One of the possible options I tried was to first reduce by e[1]: e.from_polynomial(f).reduce([a+b+c]).specialization({a:

One of the possible options I tried was to first reduce by e[1]:e.from_polynomial(f).reduce([a+b+c]).specialization({a:0

Thank you for your time and effort but I still don't see what you mean. I must be missing some piece of theory you could

Now I don't follow :-) To recover $e_i$ you surely need some original (non-primed) $e$ on the right hand side. Otherwise

Ok. It is symmetric but is there any relation between this new polynomial in 3 variables to the original one? In particu

you can reduce your polynomial in the ideal generated by polynomials like e1 (known to be zero) before converting to

If you work with symmetric expressions of 5 roots only, how do you get ei with i>5 in the result of from_polynomia

I wasn't asking how to reduce the expression in elementary symmetric polynomials, but rather how to speed up the computa

Efficient reduction to elementary symmetric polynomials When using the resolvent method for solving polynomial equations

Basically I want a structure where I have both zeta^2 (or at least for display) and ability to factor in the polyring.

Cyclotomic fields - displaying powers of zeta Is there any way how to avoid expressing $\zeta^{n-1}$ in terms of lower p

I've found out that for partial evaluation of a polynomial, there is the SpecializationMorphism. This works for whicheve

Thanks. While the idea is clear and correct, for future readers I should just note that it's better to precompute the so

Thanks. While the idea is clear and correct, for future readers I should just note that's better to precompute the sorte

I'm studying the Lagrange resolvent method for solving n-degree equations. This is what I have so far:

# a,b,c,d represent the roots of a 4-degree polynomial
R.<a,b,c,d> = QQ[]

# w will be treated as a primitive 4th root of unity
K.<w> = R[]

F = a + b*w + c*w^2 + d*w^3

# Lagrange resolvent
W = F^4 % (w^4 - 1)

# list of coefficients of w^i in W
Ws = [W[i] for i in range(4)]

S4 = SymmetricGroup(R.gens())

# compute the orbit of W under the action of S4
WOrbit = set([tuple([ws(s(R.gens())) for ws in Ws]) for s in S4])

# pretty print the orbit
for indt, t in enumerate(WOrbit):
    print(f"Res {indt + 1}:")
    for ind, i in enumerate(t):
        show(html(f"${latex(w^ind)}$: ${latex(i)}$"))

This works quite nicely except for the term ordering of the coefficients (polynomials in the roots a,b,c,d). For example, for two elements of the orbit, I have the following constant terms: $1: a^{4} + b^{4} + 12 a^{2} b c + 6 b^{2} c^{2} + c^{4} + 12 a b^{2} d + 12 a c^{2} d + 6 a^{2} d^{2} + 12 b c d^{2} + d^{4}$ $1: a^{4} + 6 a^{2} b^{2} + b^{4} + 12 a b c^{2} + c^{4} + 12 a^{2} c d + 12 b^{2} c d + 12 a b d^{2} + 6 c^{2} d^{2} + d^{4}$

I would like some consistent ordering of the terms. I guess the inconsistency is caused by permuting the variables. Is there any way to adapt the term ordering according to the used permutation?

Also, is it possible to achieve an ordering first by the "degree signature" and then lexicographically? E.g. the first example should be ordered as (I grouped the terms with the same degree signature): $1: (a^{4} + b^{4} + c^{4} + d^{4}) + (6 a^{2} d^{2} + 6 b^{2} c^{2}) + (12 a^{2} b c + 12 a b^{2} d + 12 a c^{2} d + 12 b c d^{2} )$

Last but not least, since I'm a beginner in Sage I would appreciate any comments on my solution. I'm sure there must be nicer or more efficient solutions for some of my steps.

Specific term order of polynomials I'm studying the Lagrange resolvent method for solving n-degree equations. This is wh

I've learned one thing from your alternate solution which is useful in general: That defining the target poly ring incre

How to prevent expanding the value of the generating element in symbolic expressions?

E.<w> = CyclotomicField(3)
w^4 # shows w
(w+1)^2 # shows w
a = var('a')
(w + a)^2 # shows an expression with w expanded as a complex number

How can I make the last expression show up as $w^2 + 2aw + a^2$?

My use case is to expand $(a+bw+cw^2)^3$ so that the result is expressed as a rational (or integer) combination of 1, w, w^2.

a, b = var('a,b')
((a + b)^2).expand()

How can I compute/expand the last expression assuming that $ab=0$? I.e. I would like the result to be $a^2+b^2$. I tried with assuming(..) but that doesn't have any effect.

That's great! The only downside to this approach is the false symmetry between w,a,b,c in the poly ring definition. This

That's great! The only downside to this approach is the false symmetry between w,a,b,c in the poly ring definition. This

I've enhanced my original question with the real use case.

Symbolic arithmetic in a number field How to prevent expanding the value of the generating element in symbolic expressio

Unfortunately this works only in the simplest case. Anything non-trivial, e,g, solving [a-x^2 == 0] and then converting

Thanks! How can I transfer the "polynomialness" of the parameters also to the solutions of equations? E.g. R.<a> =

Thanks, makes sense. What if the base field is more complicated, though? E.g. E.<w> = CyclotomicField(3) and I wou

Thanks, makes sense. What if the base field is more complicated, though? E.g. E.<w> = CyclotomicField(3) and I wou

Thanks, makes sense. What if the base field is more complicated, though? E.g. E.<w> = CyclotomicField(3) and I wou

Assumptions on symbolic expressions a, b = var('a,b') ((a + b)^2).expand() How can I compute/expand the last expressio

Symbolic arithmetic in a number field How to prevent expanding the value of the generating element in symbolic expressio