# solveSystem -- solutions to a system of equalities

## Synopsis

• Usage:
solveSystem(I)
• Inputs:
• I, a zero dimensional ideal.
• Optional inputs:
• Tolerance => ..., default value .001, numerical tolerance.
• Outputs:
• S, containing the solutions with their multiplicities

## Description

 i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : I = ideal(x+3*y^2-2*z, x^2-2*y-z, 3*x-4*y+5*z^2) 2 2 2 o2 = ideal (3y + x - 2z, x - 2y - z, 5z + 3x - 4y) o2 : Ideal of R i3 : M = solveSystem(I) o3 = Tally{{-.286867+1.04128*ii, -.106152-.664903*ii, -.789675+.732385*ii} => {-.286867-1.04128*ii, -.106152+.664903*ii, -.789675-.732385*ii} => {-2.02148, 1.29357, 1.49925} => 1 {.2419+1.1525*ii, -.675624+.694806*ii, .081518-.832036*ii} => 1 {.2419-1.1525*ii, -.675624-.694806*ii, .081518+.832036*ii} => 1 {0, 0, 0} => 1 {1.05571+.941279*ii, .134991+.63069*ii, -.041468+.726052*ii} => 1 {1.05571-.941279*ii, .134991-.63069*ii, -.041468-.726052*ii} => 1 ------------------------------------------------------------------------ 1} 1 o3 : Tally

## Caveat

the procedure involves computation over inexact fields. If numerical errors occur, then the multiplicities of the solutions may be not reliable.

## Ways to use solveSystem :

• "solveSystem(Ideal)"

## For the programmer

The object solveSystem is .