# degreeOfMap -- computes the degree of a rational map

## Synopsis

• Usage:
degreeOfMap(I)
• Inputs:
• I, an ideal, an ideal defining the map
• Optional inputs:
• Strategy => ..., default value Hm1Rees0Strategy, Choose a strategy for computing the degree of map
• Outputs:
• an integer, the degree of the corresponding rational map

## Description

Let $R$ be the polynomial ring $R=k[x_0,...,x_r]$ and $I$ be the homogeneous ideal $I=(f_0,f_1,...,f_s)$ where $deg(f_i)=d$. We compute the degree of the rational map $\mathbb{F}: \mathbb{P}^r \to \mathbb{P}^s$ defined by $$(x_0: ... :x_r) \to (f_0(x_0,...,x_r), f_1(x_0,...,x_r), ..... , f_s(x_0,...,x_r)).$$ The degree can be computed by two different strategies and the default one is "Hm1Rees0Strategy".

The following example is a rational map without base points:

 i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : I = ideal(random(4, R), random(4, R), random(4, R)); o2 : Ideal of R i3 : betti res I 0 1 2 3 o3 = total: 1 3 3 1 0: 1 . . . 1: . . . . 2: . . . . 3: . 3 . . 4: . . . . 5: . . . . 6: . . 3 . 7: . . . . 8: . . . . 9: . . . 1 o3 : BettiTally i4 : degreeOfMap I o4 = 16

In the following examples we play with the relations of the Hilbert-Burch presentation and the degree of $\mathbb{F}$ (see Proposition 5.2 and Theorem 5.12):

 i5 : A = matrix{ {x, x^2 + y^2}, {-y, y^2 + z*x}, {0, x^2} }; 3 2 o5 : Matrix R <--- R i6 : I = minors(2, A) -- a birational map 2 2 3 2 3 2 o6 = ideal (x y + x*y + y + x z, x , -x y) o6 : Ideal of R i7 : degreeOfMap I o7 = 1 i8 : A = matrix{ {x^2, x^2 + y^2}, {-y^2, y^2 + z*x}, {0, x^2} }; 3 2 o8 : Matrix R <--- R i9 : I = minors(2, A) -- a non birational map 2 2 4 3 4 2 2 o9 = ideal (2x y + y + x z, x , -x y ) o9 : Ideal of R i10 : degreeOfMap I o10 = 2 i11 : A = matrix{ {x^3, x^2 + y^2}, {-y^3, y^2 + z*x}, {0, x^2} }; 3 2 o11 : Matrix R <--- R i12 : I = minors(2, A) -- a non birational map 3 2 2 3 5 4 5 2 3 o12 = ideal (x y + x y + y + x z, x , -x y ) o12 : Ideal of R i13 : degreeOfMap I o13 = 3 i14 : A = matrix{ {x^3, x^4}, {-y^3, y^4}, {z^3, x^4} }; 3 2 o14 : Matrix R <--- R i15 : I = minors(2, A) -- a non birational map 4 3 3 4 7 4 3 4 3 4 3 o15 = ideal (x y + x y , x - x z , - x y - y z ) o15 : Ideal of R i16 : degreeOfMap I o16 = 12

The following examples are computed with the strategy "SatSpecialFibStrategy".

 i17 : R = QQ[x,y,z,v,w] o17 = R o17 : PolynomialRing i18 : I = ideal(random(1, R), random(1, R), random(1, R), random(1, R), random(1, R)); o18 : Ideal of R i19 : degreeOfMap(I, Strategy=>SatSpecialFibStrategy) o19 = 1 i20 : I = ideal(29*x^3 + 55*x*y*z, 7*y^3, 14*z^3, 17*v^3, 12*w^3) 3 3 3 3 3 o20 = ideal (29x + 55x*y*z, 7y , 14z , 17v , 12w ) o20 : Ideal of R i21 : degreeOfMap(I, Strategy=>SatSpecialFibStrategy) o21 = 81

## Caveat

To call the method "degreeOfMap(I)", the ideal $I$ should be in a single graded polynomial ring.

## Ways to use degreeOfMap :

• "degreeOfMap(Ideal)"

## For the programmer

The object degreeOfMap is .