Pol

Function: Pol
Section: conversions
C-Name: gtopoly
Prototype: GDn
Description:
 (gen,?var):pol  gtopoly($1, $2)
Help: Pol(t,{v='x}): convert t (usually a vector or a power series) into a
 polynomial with variable v, starting with the leading coefficient.
Doc:
 transforms the object $t$ into a polynomial with main variable $v$. If $t$
 is a scalar, this gives a constant polynomial. If $t$ is a power series with
 nonnegative valuation or a rational function, the effect is similar to
 \kbd{truncate}, i.e.~we chop off the $O(X^k)$ or compute the Euclidean
 quotient of the numerator by the denominator, then change the main variable
 of the result to $v$.

 The main use of this function is when $t$ is a vector: it creates the
 polynomial whose coefficients are given by $t$, with $t[1]$ being the leading
 coefficient (which can be zero). It is much faster to evaluate
 \kbd{Pol} on a vector of coefficients in this way, than the corresponding
 formal expression $a_n X^n + \dots + a_0$, which is evaluated naively exactly
 as written (linear versus quadratic time in $n$). \tet{Polrev} can be used if
 one wants $x[1]$ to be the constant coefficient:
 \bprog
 ? Pol([1,2,3])
 %1 = x^2 + 2*x + 3
 ? Polrev([1,2,3])
 %2 = 3*x^2 + 2*x + 1
 @eprog\noindent
 The reciprocal function of \kbd{Pol} (resp.~\kbd{Polrev}) is \kbd{Vec} (resp.~
 \kbd{Vecrev}).
 \bprog
 ? Vec(Pol([1,2,3]))
 %1 = [1, 2, 3]
 ? Vecrev( Polrev([1,2,3]) )
 %2 = [1, 2, 3]
 @eprog\noindent

 \misctitle{Warning} This is \emph{not} a substitution function. It will not
 transform an object containing variables of higher priority than~$v$.
 \bprog
 ? Pol(x + y, y)
   ***   at top-level: Pol(x+y,y)
   ***                 ^----------
   *** Pol: variable must have higher priority in gtopoly.
 @eprog