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This is probably an easy question, but I have been looking for hours before posting it. I want for a given finite set say $S$ (in particular a set of homogeneous polynomials), to define the vector space over a finite field $\mathbb{F}$ spanned from that set. I wasn't able to find out online how to create a non-standard vector space $V$; a standard vector space (I use magma's terminology) is defined to be $\mathbb{F}^n$. Moreover, I can't find out how to define the vector space of the homogeneous component of degree $d$, say $R^{(d)}$ for a given polynomial algebra, $R:=\mathbb{F}[x_1,...,x_n]$. Any reference is welcomed.

P.S. My question may be a naive one, though I've spent much time looking for it. Any sort of help is appreciated since I really need that command to proceed.

user430191
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  • I want to define a function that checks the membership of a polynomial inside the homogeneous of degree-d polynomials in a polynomial algebra, after we have applied a twisted derivation on it. In order to do this, I need firstly to define the d-th degree homogeneous component of this polynomial algebra. If you have any idea how this can be done more easily I can provide further information. – user430191 May 20 '18 at 14:38
  • Hmm, I would try a formal set construction maybe? Something like Hd := {! p in P | {Degree(f) : f in Monomials(p)} eq {d} !} – xxxxxxxxx May 20 '18 at 16:33
  • @MorganRodgers thank you very much, this is helpful indeed since I am a primer on magma. However, I am still struggling to write out the code for my initial query. I mean, I would like to write out a set which consists of polynomials, after a twisted derivation has been applied. In particular, I have an action of a cyclic group $G=<\sigma>$ on a polynomial algebra say $R$, and the twisted derivation $\Delta=\sigma-1$ acts on $R$. Do you know how can I define the subset of R which is the $\Delta(R):={ \Delta(f) | f \in R}$ in MAGMA? After that I want to check whether a polynomial lies in it. – user430191 May 21 '18 at 11:37

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