-*- M2 -*-
Title: Solving Equations
Description:
Solving equations can be done with Groebner bases, provided we choose a
monomial ordering adapted to elimination of variables (not the default
ordering). Engineers and others could use Macaulay2 if it could solve
equations.
Here is a simple example:
i4 : R = QQ[x,y,MonomialOrder=>Lex]
o4 = R
o4 : PolynomialRing
i7 : compactMatrixForm = false
o7 = false
i9 : I = ideal ( x^2-y^2+11+y, x^3 - x*y^2 +12 )
2 2 3 2
o9 = ideal (x - y + y + 11, x - x*y + 12)
o9 : Ideal of R
i10 : gens gb I
| 4 3 2 3 2 |
o10 = | y + 21y + 88y - 363y - 1475 12x - y - 10y + 22y + 121 |
1 2
o10 : Matrix R <--- R
i11 : transpose oo
| 4 3 2 |
o11 = | y + 21y + 88y - 363y - 1475 |
| |
| 3 2 |
| 12x - y - 10y + 22y + 121 |
2 1
o11 : Matrix R <--- R
The new system of equations in line o11 is simpler than the original system,
because it is "triangular". The first equation is involves just y and is monic
in y, with 4 roots. The second equation involves x and y, and as a polynomial
in x with coefficients that are polynomials in y, it is monic (of degree 1 in
x). We conclude easily that the system has 4 solutions, which could be found
numerically, if desired.
Not all gb's lead immediately to triangular systems, and the main problem with
using gb's is that they easily suffer a combinatorial explosion that leads to
unusable answers. The main tool in that case it to compute for a while (our gb
function will stop after various criteria are satisfied and provide the gb
generators it has found so far). Examination of the generators will often
suggest an auxiliary polynomial g, which, if only it were invertible modulo I,
woulc lead to the discovery of a monic equation in the frirst (next) variable.
For example, we may have an equation h=h(x,y) including a g = g(y) times a high
(highest) power of x.
Having chosen g, we observe that at every solution P of our system has either
g(P)=0 or g(P)!=0. So, it is enough to solve two new systems: one where g=0 is
appended to the list of equations, and one where g!=0 is appended (actually, we
add a new variable u and append a new equation ug=1.) In the system with g=0
our equation h reduces its degree in x. In the system with ug=1, provided we
choose the new monomial ordering properly, the next gb computation will quickly
replace h by something monic in x by trading it off against the equation ug=1,
and that will hasten total progress.
Here is an example. We consider the generic linear equation in two variables,
x and y. We choose an elmination ordering where the variables are preferred to
the coefficients.
i10 : R = ZZ[a,b,c,d,p,q][x,y]
o10 = R
o10 : PolynomialRing
i11 : I = ideal(a*x+b*y-p,c*x+d*y-q)
o11 = ideal (a*x + b*y - p, c*x + d*y - q)
o11 : Ideal of R
i15 : transpose gens gb I
o15 = | (b*c - a*d)y - c*p + a*q |
| |
| c*x + d*y - q |
| |
| a*x + b*y - p |
3 1
o15 : Matrix R <--- R
Notice that the coefficient of y in the top equation is the determinant, and
inverting it leads to a monic linear equation in y, i.e., a formula for y.
Here is an example involving the discriminant of a polynomial.
i16 : R = ZZ[p,q][x]
o16 = R
o16 : PolynomialRing
i17 : f = x^3 + p*x + q
3
o17 = x + p*x + q
o17 : R
i19 : I = ideal(f, diff_x f)
3 2
o19 = ideal (x + p*x + q, 3x + p)
o19 : Ideal of R
i20 : transpose gens gb I
| 3 2 |
o20 = | 4p + 27q |
| |
| 2 |
| 9q*x - 2p |
| |
| 2p*x + 3q |
| |
| 3 2 |
| p*q*x + 2p + 15q |
| |
| 2 |
| 3x + p |
| |
| 2 2 |
| p*x - 3q*x + p |
| |
| 3 |
| x + p*x + q |
7 1
o20 : Matrix R <--- R
Notice that the top equation (generator) does not involve x. Let's set it
equal to zero and see what happens:
i24 : S = (ZZ[p,q]/(4*p^3+27*q^2))[x]
o24 = S
o24 : PolynomialRing
i25 : f = x^3 + p*x + q
3
o25 = x + p*x + q
o25 : S
i26 : J = ideal(f, diff_x f)
3 2
o26 = ideal (x + p*x + q, 3x + p)
o26 : Ideal of S
i27 : transpose gens gb J
| 2 |
o27 = | 9q*x - 2p |
| |
| 2p*x + 3q |
| |
| 3 2 |
| p*q*x + 2p + 15q |
| |
| 2 |
| 3x + p |
| |
| 2 2 |
| p*x - 3q*x + p |
| |
| 3 |
| x + p*x + q |
6 1
o27 : Matrix S <--- S
We see from the top equation that inverting q and 3 would give a monic equation
for x, and from the second equation that inverting p and 2 would, too. And so
on.
Alternatively, we may invert the discriminant:
i41 : T = (ZZ[p,q,u]/(u*(4*p^3+27*q^2)-1))[x]
o41 = T
o41 : PolynomialRing
i42 : f = x^3 + p*x + q
3
o42 = x + p*x + q
o42 : T
i43 : J = ideal(f, diff_x f)
3 2
o43 = ideal (x + p*x + q, 3x + p)
o43 : Ideal of T
i44 : transpose gens gb J
o44 = | 1 |
1 1
o44 : Matrix T <--- T
We get the equation 1==0, which has no solutions. Hence, when the disriminant
is invertible, f(x)=f'(x)==0 has no solutions, a standard fact.
Here is an example involving good and bad reduction of elliptic curves. We
take an (affine) elliptic curve f==0 and seek its singular points.
i45 : R = ZZ[x,y]
o45 = R
o45 : PolynomialRing
i54 : f = y^2 + y - x^3 + x^2
3 2 2
o54 = - x + x + y + y
o54 : R
i55 : transpose gens gb ideal(f, diff_x f, diff_y f)
o55 = | 11 |
| |
| y - 5 |
| |
| x + 3 |
3 1
o55 : Matrix R <--- R
The system is completely solved: there is one singular point at x=-3 y=5 11=0.
The number 11 is a prime of bad reduction for the curve.
A good heuristic algorithm can be developed by experimentation with
computations such as those above.
=============================================================================
Proposed by: Dan Graysonn
Potential Advisor: Dan Graysonn
Project assigned to:
Current status:
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Progress log: