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tutorial:polynomials_tutorial [2011/08/01 13:10] – [Example: Toric Degeneration] gawrilowuser_guide:tutorials:polynomials_tutorial [2019/02/04 22:55] (current) – external edit 127.0.0.1
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-===== Basic Usage of Polynomials in Perl ===== +{{page>.:latest:@FILEID@}}
- +
-A polynomial always carries a reference to a polynomial ring.  Polynomial in different rings cannot be added, multiplied, or whatever.  The names of the indeterminates are relevant for the output functions only. +
- +
-<code> +
-$r=new Ring(qw(x y)); +
-</code> +
- +
-It may be convenient to use special variable names for the indeterminates. +
- +
-<code> +
-polytope> ($x,$y)=$r->variables; +
-</code> +
- +
-Notice that the variable names are chosen to match the print names; but this is "coincidental". +
- +
-<code> +
-polytope> $p=1+$x+$y; +
-polytope> $q=1+$x+$y+$x*$y; +
-polytope> print($p*$q); +
-1 + 2*x + x^2 + 2*y + 3*x*y + y^2 + x^2*y + x*y^2 +
-</code> +
- +
-Standard arithmetic function "+", "-", "*", "^" are defined.  However, due to the fact that their precedence is given in perl it may be necessary to write more parentheses than expected at first sight. +
- +
-Here is one way to produce polytopes from polynomials (as the convex hull of the exponent vectors of all terms). +
- +
-<code> +
-polytope> $npq=newton($p*$q); +
-polytope> print $npq->VERTICES; +
-1 0 0 +
-1 2 0 +
-1 0 2 +
-1 2 1 +
-1 1 2 +
-polytope> print equal_polyhedra($npq, minkowski_sum(newton($p),newton($q))); +
-1 +
-</code> +
- +
-The final "1" means "true": The Newton polytope of the product of two polynomials always equals the Minkowski sum of the Newton polytopes of the factors. +
- +
-===== Example: Toric Degeneration ===== +
- +
-The following describes how to construct the polynomial which describes the toric deformation with respect to a point configuration and a height function.  This is the input data: +
- +
-<code> +
-polytope > $points = new Matrix<Int>([1,0],[0,1]); +
-polytope > $height = new Vector<Int>([2,3]); +
-polytope > $coefficients = new Vector<Rational>([-1/2,1/3]); +
-</code> +
- +
-The following is generic (assuming that the dimensions of the objects above match). +
- +
-<code> +
-polytope > @vars = "s"; push @vars, map { "x$_" } (1..$points->rows()); +
-polytope > $R = new Ring(@vars); +
-polytope > $p = new Polynomial($height|$points,$coefficients,$R); +
-</code> +
- +
-Notice that the points are given in Euclidean coordinates; that is, if applied, e.g., to the VERTICES of a polytope do not forget to strip the homogenizing coordinate. The output in our example looks like this: +
- +
-<code> +
-polytope > print $p; +
--1/2*s^2*x1 + 1/3*s^3*x2 +
-</code> +
  
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