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Involutive bases are a new tool for solving problems in connection with multivariate polynomials, such as solving systems of polynomial equations and analyzing polynomial ideals. An involutive basis of polynomial ideal is nothing but a special form of a redundant Gröbner basis. The construction of involutive bases reduces the problem of solving polynomial systems to simple linear algebra.
Authors: A.Yu. Zharkov and Yu.A. Blinkov.
Involutive bases are a new tool for solving problems in connection with multivariate
polynomials, such as solving systems of polynomial equations and analyzing polynomial
ideals, see [ZB96]. An involutive basis of polynomial ideal is nothing but a special form
of a redundant Gröbner basis. The construction of involutive bases reduces the problem
of solving polynomial systems to simple linear algebra.
The INVBASE package 26
calculates involutive bases of polynomial ideals using an algorithm described in [ZB96]
which may be considered as an alternative to the well-known Buchberger algorithm
[Buc85]. The package can be used over a variety of different coefficient domains, and for
different variable and term orderings.
The algorithm implemented in the INVBASE package is proved to be valid for any
zero-dimensional ideal (finite number of solutions) as well as for positive-dimensional
ideals in generic form. However, the algorithm does not terminate for “sparse”
positive-dimensional systems. In order to stop the process we use the maximum degree
bound for the Gröbner bases of generic ideals in the total-degree term ordering
established in [Laz83]. In this case, it is reasonable to call the GROEBNER package with
the answer of INVBASE as input information in order to compute the reduced Gröbner
basis under the same variable and term ordering.
Though the INVBASE package supports computing involutive bases in any admissible
term ordering, it is reasonable to compute them only for the total-degree term
orderings. The package includes a special algorithm for conversion of total-degree
involutive bases into the triangular bases in the lexicographical term ordering
that is desirable for finding solutions. Normally the sum of timings for these
two computations is much less than the timing for direct computation of the
lexicographical involutive bases. As a rule, the result of the conversion algorithm
is a reduced Gröbner basis in the lexicographical term ordering. However,
because of some gaps in the current version of the algorithm, there may be rare
situations when the resulting triangular set does not possess the formal property
of Gröbner bases. Anyway, we recommend using the GROEBNER package
with the result of the conversion algorithm as input in order either to check the
Gröbner bases property or to transform the result into a lexicographical Gröbner
basis.
The following term order modes are available:
revgradlex; gradlex; lex .
These modes have the same meaning as for the GROEBNER package.
All orderings are based on an ordering among the variables. For each pair of variables an
order relation \(>\) must be defined, e.g. \(x>y\). The term ordering mode as well as the order of
variables are set by the operator
where \(\langle \)mode\(\rangle \) is one of the term order modes listed above. The notion of {\(x_1\),…,\(x_n\)} as a list of
variables at the same time means \(x_1>...>x_n\).
Example 1.
invtorder revgradlex,{x,y,z}
sets the reverse graduated term ordering based on the variable order \(x>y>z\).
The operator invtorder may be omitted. The default term order mode is
revgradlex and the default decreasing variable order is alphabetical (or, more
generally, the default REDUCE kernel order). Furthermore, the list of variables
in the invtorder may be omitted. In this case the default variable order is
used.
To compute the involutive basis of ideal generated by the set of polynomials \(\{p_1,...,p_m\}\) one should type the command
where \(p_i\) are polynomials in variables listed in the invtorder operator. If some kernels
in \(p_i\) were not listed previously in the invtorder operator they are considered as
parameters, i.e. they are considered part of the coefficients of polynomials. If
invtorder was omitted, all the kernels in \(p_i\) are considered as variables with the default
REDUCE kernel order.
The coefficients of polynomials \(p_i\) may be integers as well as rational numbers (or,
accordingly, polynomials and rational functions in the parametric case). The
computations modulo prime numbers are also available. For this purpose one should type
the REDUCE commands
on modular; setmod p;
where \(p\) is a prime number.
The value of the invbase function is a list of integer polynomials \(\{g_1,...,g_n\}\) representing an
involutive basis of a given ideal.
Example 2.
invtorder revgradlex,{x,y,z};
g:= invbase {4*x**2 + x*y**2 - z + 1/4,
2*x + y**2*z + 1/2,
x**2*z - 1/2*x - y**2 };
The resulting involutive basis in the reverse graduate ordering is
3 2 3 2
g := {8*x*y*z - 2*x*y*z + 4*y - 4*y*z + 16*x*y
+ 17*y*z - 4*y,
4 2 2 2
8*y - 8*x*z - 256*y + 2*x*z + 64*z
- 96*x + 20*z - 9,
3
2*y *z + 4*x*y + y,
3 2 2 2
8*x*z - 2*x*z + 4*y - 4*z + 16*x + 17*z
- 4,
3 3 2
- 4*y*z - 8*y + 6*x*y*z + y*z - 36*x*y
- 8*y,
2 2 2
4*x*y + 32*y - 8*z + 12*x - 2*z + 1,
2
2*y *z + 4*x + 1,
3 2 2
- 4*z - 8*y + 6*x*z + z - 36*x - 8,
2 2 2
8*x - 16*y + 4*z - 6*x - z}
To convert it into a lexicographical Gröbner basis one should type
h:=invlex g;
The result is
6 5 4
h := {3976*x + 37104*z - 600*z + 2111*z
3 2
+ 122062*z + 232833*z - 680336*z + 288814
,
2 6 5 4
1988*y - 76752*z + 1272*z - 4197*z
3 2
- 251555*z - 481837*z + 1407741*z
- 595666,
7 6 5 4 3 2
16*z - 8*z + z + 52*z + 75*z - 342*z
+ 266*z - 60}
In the case of “sparse” positive-dimensioned system when the involutive basis in the sense of [ZB96] does not exist, you get the error message
***** Maximum degree bound exceeded.
The resulting list of polynomials which is not an involutive basis is stored in the share variable invtempbasis. In this case it is reasonable to call the GROEBNER package with the value of invtempbasis as input under the same variable and term ordering.
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