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# upperBoundDegreeSingleGraded -- computes an upper bound for the degree of a rational map

## Synopsis

• Usage:
upperBoundDegreeSingleGraded(I)
• Inputs:
• I, an ideal, an ideal defining the map
• Outputs:
• an integer, an upper bound for 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 $$(x_0: ... :x_r) \to (f_0(x_0,...,x_r), f_1(x_0,...,x_r), ..... , f_s(x_0,...,x_r)).$$ Using certain Hilbert functions the degree of the map is bounded (see Theorem 3.22 in Degree and birationality of multi-graded rational maps).

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 i5 : upperBoundDegreeSingleGraded I o5 = 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):

 i6 : A = matrix{ {x, x^2 + y^2}, {-y, y^2 + z*x}, {0, x^2} }; 3 2 o6 : Matrix R <-- R i7 : I = minors(2, A) -- a birational map 2 2 3 2 3 2 o7 = ideal (x y + x*y + y + x z, x , -x y) o7 : Ideal of R i8 : degreeOfMap I o8 = 1 i9 : upperBoundDegreeSingleGraded I o9 = 2 i10 : A = matrix{ {x^2, x^2 + y^2}, {-y^2, y^2 + z*x}, {0, x^2} }; 3 2 o10 : Matrix R <-- R i11 : I = minors(2, A) -- a non birational map 2 2 4 3 4 2 2 o11 = ideal (2x y + y + x z, x , -x y ) o11 : Ideal of R i12 : degreeOfMap I o12 = 2 i13 : upperBoundDegreeSingleGraded I o13 = 4 i14 : A = matrix{ {x^3, x^2 + y^2}, {-y^3, y^2 + z*x}, {0, x^2} }; 3 2 o14 : Matrix R <-- R i15 : I = minors(2, A) -- a non birational map 3 2 2 3 5 4 5 2 3 o15 = ideal (x y + x y + y + x z, x , -x y ) o15 : Ideal of R i16 : degreeOfMap I o16 = 3 i17 : upperBoundDegreeSingleGraded I o17 = 7

## Caveat

To call the method "degreeOfMap(I)", the ideal $I$ should be in a single graded polynomial ring and dim(R/I) <= 1.