You can use both the Solve > Expression1PET_XA and Solve > System command1EJYXJP commands to solve systems of polynomial equations.  You can also use the SOLVE and SOLUTIONS functions to solve such systems.  The first argument of SOLVE and SOLUTIONS is the system of equations to be solved.  The second argument is a vector of the solution variables.


All these methods use the Gröbner BasisGROEBNER_BASIS to reduce the system of polynomial equations to a single univariate equation.  This univariate equation is then solved algebraically or numerically, and the result back-substituted into the other equations.


Given a conjunction of equations, the Solve > Expression command1PET_XA and the SOLVE function return the solutions as a Boolean expression logically equivalent to the input but in disjunctive normal form.  That is an expression of the form

conjunct 1 OR conjunct 2 OR ... 

where conjunct i is an expression of the form

relation i1 AND relation i2 AND ...

where relation ij is an equation or inequality.  For example,

SOLVE(x^2 + y^2 = 2·a^2 AND x - y = 0, [x, y])

simplifies to

(x = -a ∧ y = -a) ∨ (x = a ∧ y = a)


Given a vector of equations, the Solve > Expression command1PET_XA and the SOLVE function return a vector of solutions.  The Solve System command1EJYXJP always returns the solutions in this form.  For example, 

SOLVE([x^2 + y^2 = 2·a^2, x - y = 0], [x, y])

simplifies to

[x = a ∧ y = a, x = -a ∧ y = -a]


No matter how the equations are given, SOLUTIONS returns a vector of vectors (i.e. a matrix) of solution values.  For example,

SOLUTIONS(x^2 + y^2 = 2·a^2 AND x - y = 0, [x, y])

simplifies to

⎡  a   a ⎤
⎢        ⎥
⎣ -a  -a ⎦


A system of linear equations is singular if the equations are not independent of one another.  Singular systems of linear equations can be either consistent or inconsistent.


Given a consistent singular system of linear equations, both the Solve commandsSolve_commands and the SOLVE function return a simplified equivalent system of equations.  For example,

SOLVE(2·x + 6·y = 2 AND 3·x = 3 - 9·y, [x, y])

simplifies to

x + 3·y = 1


Given a consistent singular system of linear equations, the SOLUTIONS function returns a vector of solutions containing one or more arbitrary values of the form @n where n is an integer.  The number of distinct arbitrary values is equal to the number of solution variables minus the number of independent equations.   For example,

SOLUTIONS(2·x + 6·y = 2 AND 3·x = 3 - 9·y, [x, y])

simplifies to

⎡      1 - @1  ⎤
⎢ @1  ———————— ⎥
⎣         3    ⎦


Given an inconsistent singular system of linear equations, the Solve commandsSolve_commands and the SOLVE function return the truth-value false or the empty vector [ ], depending on how the system is given.  For example,

SOLVE(x + 3·y = 1 AND 2·x + 6·y = 3, [x, y])

simplifies to false and

SOLVE([x + 3·y = 1, 2·x + 6·y = 3], [x, y])

simplifies to [ ].


The Solve commandsSolve_commands and the SOLVE and SOLUTIONS functions cannot directly solve a system of equations if they are not polynomial in their solution variables.  Functions are defined in the EquationSolving.mth.S02QQ utility file for computing approximate real or complex solutions to systems of non-polynomial equations.

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