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--- tutorial:polynomials_tutorial [2011/07/13 08:57] – [Example: Toric Degeneration] joswig
+++ user_guide:tutorials:polynomials_tutorial [2019/02/04 22:55] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
-===== 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);
-+ 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 $npq->VERTICES;
-0 0
-2 0
-0 2
-2 1
-1 2
-polytope> print equal_polyhedra($npq, minkowski_sum(newton($p),newton($q)));
-</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 > $coefficients = new Vector<Rational>([-1/2,1/3]);
-</code>
-The following is generic (assuming that the dimensions of the objects above match).
-<code>
-polytope > @vars = "s"; map { push @vars, "x$_" } (1..$points->rows());
-polytope > $R = new Ring(@vars);
-polytope > $p=new Polynomial($height|$points,$coefficients,$R);
-polytope > print $p;
--1/2*s^2*x1 + 1/3*s^3*x2
-</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.