# noetherNormalization -- data for Noether normalization

## Synopsis

• Usage:
(f,J,X) = noetherNormalization C
• Inputs:
• Optional inputs:
• LimitList => a list, default value {5, 20, 40, 60, 80, infinity}, gives the value which
• RandomRange => an integer, default value 0, if not 0, gives a integer bound for the random coefficients. If 0, then chooses random elements from the coefficient field.
• Verbose => ..., default value false
• Outputs:
• f, , an automorphism of R
• J, an ideal, the image of I under f
• X, a list, a list of variables which are algebraically independent in R/J

## Description

The computations performed in the routine noetherNormalization use a random linear change of coordinates, hence one should expect the output to change each time the routine is executed.
 i1 : R = QQ[x_1..x_4]; i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1); o2 : Ideal of R i3 : (f,J,X) = noetherNormalization I 9 9 1 1 13 2 o3 = (map (R, R, {-x + -x + x , x , -x + -x + x , x }), ideal (--x + 4 1 2 2 4 1 2 1 2 2 3 2 4 1 ------------------------------------------------------------------------ 9 9 3 27 2 2 9 3 9 2 9 2 1 2 -x x + x x + 1, -x x + --x x + -x x + -x x x + -x x x + -x x x + 2 1 2 1 4 8 1 2 8 1 2 4 1 2 4 1 2 3 2 1 2 3 2 1 2 4 ------------------------------------------------------------------------ 1 2 -x x x + x x x x + 1), {x , x }) 2 1 2 4 1 2 3 4 4 3 o3 : Sequence
The next example shows how when we use the lexicographical ordering, we can see the integrality of R/ f I over the polynomial ring in dim(R/I) variables:
 i4 : R = QQ[x_1..x_5, MonomialOrder => Lex]; i5 : I = ideal(x_2*x_1-x_5^3, x_5*x_1^3); o5 : Ideal of R i6 : (f,J,X) = noetherNormalization I 3 7 3 7 3 o6 = (map (R, R, {-x + x + x , x , -x + -x + x , -x + -x + x , x }), 4 1 2 5 1 4 1 4 2 4 9 1 2 2 3 2 ------------------------------------------------------------------------ 3 2 3 27 3 27 2 2 27 2 9 3 ideal (-x + x x + x x - x , --x x + --x x + --x x x + -x x + 4 1 1 2 1 5 2 64 1 2 16 1 2 16 1 2 5 4 1 2 ------------------------------------------------------------------------ 9 2 9 2 4 3 2 2 3 -x x x + -x x x + x + 3x x + 3x x + x x ), {x , x , x }) 2 1 2 5 4 1 2 5 2 2 5 2 5 2 5 5 4 3 o6 : Sequence i7 : transpose gens gb J o7 = {-10} | x_2^10 {-10} | 12x_1x_2x_5^6-54x_2^9x_5-12x_2^9+27x_2^8x_5^2+12x_2^8x_5-9x_2^7x {-9} | 48x_1x_2^2x_5^3-108x_1x_2x_5^5+48x_1x_2x_5^4+486x_2^9-243x_2^8x_ {-9} | 384x_1x_2^3+864x_1x_2^2x_5^2+768x_1x_2^2x_5+2916x_1x_2x_5^5-648x {-3} | 3x_1^2+4x_1x_2+4x_1x_5-4x_2^3 ------------------------------------------------------------------------ _5^3-12x_2^7x_5^2+12x_2^6x_5^3-12x_2^5x_5^4+12x_2^4x_5^5+16x_2 5-36x_2^8+81x_2^7x_5^2+72x_2^7x_5-108x_2^6x_5^2+108x_2^5x_5^3- _1x_2x_5^4+576x_1x_2x_5^3+384x_1x_2x_5^2-13122x_2^9+6561x_2^8x ------------------------------------------------------------------------ ^2x_5^6+16x_2x_5^7 108x_2^4x_5^4+48x_2^4x_5^3+64x_2^3x_5^3-144x_2^2x_5^5+128x_2^2x_5^4-144x _5+1458x_2^8-2187x_2^7x_5^2-2430x_2^7x_5+216x_2^7+2916x_2^6x_5^2-648x_2^ ------------------------------------------------------------------------ _2x_5^6+64x_2x_5^5 6x_5-288x_2^6-2916x_2^5x_5^3+648x_2^5x_5^2+288x_2^5x_5+384x_2^5+2916x_2^ ------------------------------------------------------------------------ 4x_5^4-648x_2^4x_5^3+576x_2^4x_5^2+384x_2^4x_5+512x_2^4+1152x_2^3x_5^2+ ------------------------------------------------------------------------ 1536x_2^3x_5+3888x_2^2x_5^5-864x_2^2x_5^4+1920x_2^2x_5^3+1536x_2^2x_5^2+ ------------------------------------------------------------------------ | | | 3888x_2x_5^6-864x_2x_5^5+768x_2x_5^4+512x_2x_5^3 | | 5 1 o7 : Matrix R <--- R
If noetherNormalization is unable to place the ideal into the desired position after a few tries, the following warning is given:
 i8 : R = ZZ/2[a,b]; i9 : I = ideal(a^2*b+a*b^2+1); o9 : Ideal of R i10 : (f,J,X) = noetherNormalization I --warning: no good linear transformation found by noetherNormalization 2 2 o10 = (map (R, R, {b, a}), ideal(a b + a*b + 1), {b}) o10 : Sequence
Here is an example with the option Verbose => true:
 i11 : R = QQ[x_1..x_4]; i12 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1); o12 : Ideal of R i13 : (f,J,X) = noetherNormalization(I,Verbose => true) --trying random transformation: 1 --trying with basis element limit: 5 --trying with basis element limit: 20 7 3 7 2 o13 = (map (R, R, {7x + --x + x , x , -x + -x + x , x }), ideal (8x + 1 10 2 4 1 7 1 3 2 3 2 1 ----------------------------------------------------------------------- 7 3 499 2 2 49 3 2 7 2 --x x + x x + 1, 3x x + ---x x + --x x + 7x x x + --x x x + 10 1 2 1 4 1 2 30 1 2 30 1 2 1 2 3 10 1 2 3 ----------------------------------------------------------------------- 3 2 7 2 -x x x + -x x x + x x x x + 1), {x , x }) 7 1 2 4 3 1 2 4 1 2 3 4 4 3 o13 : Sequence
The first number in the output above gives the number of linear transformations performed by the routine while attempting to place I into the desired position. The second number tells which BasisElementLimit was used when computing the (partial) Groebner basis. By default, noetherNormalization tries to use a partial Groebner basis. It does this by sequentially computing a Groebner basis with the option BasisElementLimit set to predetermined values. The default values come from the following list:{5,20,40,60,80,infinity}. To set the values manually, use the option LimitList:
 i14 : R = QQ[x_1..x_4];  i15 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1); o15 : Ideal of R i16 : (f,J,X) = noetherNormalization(I,Verbose => true,LimitList => {5,10}) --trying random transformation: 1 --trying with basis element limit: 5 --trying with basis element limit: 10 2 5 6 5 2 o16 = (map (R, R, {-x + -x + x , x , x + -x + x , x }), ideal (-x + 3 1 2 2 4 1 1 7 2 3 2 3 1 ----------------------------------------------------------------------- 5 2 3 43 2 2 15 3 2 2 5 2 2 -x x + x x + 1, -x x + --x x + --x x + -x x x + -x x x + x x x 2 1 2 1 4 3 1 2 14 1 2 7 1 2 3 1 2 3 2 1 2 3 1 2 4 ----------------------------------------------------------------------- 6 2 + -x x x + x x x x + 1), {x , x }) 7 1 2 4 1 2 3 4 4 3 o16 : Sequence
To limit the randomness of the coefficients, use the option RandomRange. Here is an example where the coefficients of the linear transformation are random integers from -2 to 2:
 i17 : R = QQ[x_1..x_4]; i18 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1); o18 : Ideal of R i19 : (f,J,X) = noetherNormalization(I,Verbose => true,RandomRange => 2) --trying random transformation: 1 --trying with basis element limit: 5 --trying with basis element limit: 20 2 o19 = (map (R, R, {2x - x + x , x , x + x , x }), ideal (3x - x x + x x 1 2 4 1 2 3 2 1 1 2 1 4 ----------------------------------------------------------------------- 2 2 3 2 2 2 + 1, 2x x - x x + 2x x x - x x x + x x x + x x x x + 1), {x , 1 2 1 2 1 2 3 1 2 3 1 2 4 1 2 3 4 4 ----------------------------------------------------------------------- x }) 3 o19 : Sequence

This symbol is provided by the package NoetherNormalization.

## Ways to use noetherNormalization :

• "noetherNormalization(Ideal)"
• "noetherNormalization(PolynomialRing)"
• "noetherNormalization(QuotientRing)"

## For the programmer

The object noetherNormalization is .