- General

To start calculations with bergman, you are to fill the form in and to click the button [Go].

You are to fill variable and relation fields, this is obligatory.

You can use default settings for some elements. E.g., by default the mode of calculations is commutative.

Your settings are kept till reloading the page, so you can repeat calculations changing only some of them.

See the manual for detail explanations. See also
our reference page.

- Limitations

Maximum dergee is restricted by 100 for this **bergman** demo, so are Hilbert series and Anick resolution.

There are also space and time limitations. The limitations for background calculations
are less severe.

- Commutativity

Click the radio button of the desired mode: commutative or non-commutative. The most interesting part of bergman is its non-commutative calculations.

- Ordering

Click the radio button of the desired ordering: lexicographical or reverse lexicographical. Both compare monomials by their weights (degree) and then by variables' names in your given order, i.e. the algebra is graded.

- Characteristic

Enter the coefficient field characteristic. It may be 0 or a small prime number: 2,3,5,7,11,13,17,...,32761.

- Variables

This field is obligatory.

Enter variables in the desired order. This order of variables will be used during lexicographical or reverse lexicographical sorting.
E.g. if you enter 'z,x' then z became lexicographically less than x.

Variable names are case insensitive and can consist of one or several letters a..z. E.g., 'a,b,x,y,z,abc,abracadabra' are valid variable
names.

Variables are to be separated by commas. You can enter one or more names in a line. The last character of line is to be comma
if the next line contains variable. Do not put comma after the last variable.

You can not use spaces inside variable names.

If you use weights, unmark the checkbox '[x] Do not use weights' and enter weights.

Weights are positive integers and are to be given for each variable.

You can enter one or several weights in a line. Weights are to be separated by commas (like variables).

- Relations

This field is obligatory.

The realtions are to be **homogeneous**.

Enter relations using numeric coefficients, variables, and arithmetic operators +, -, *, and ^ or ** followed by the power of variable.

Relations are to be separated by commas. You can enter one or more relations in a line. The last character of line is to be comma
if the next line contains relation. Do not put comma after the last relation.

The multiplication sign * is obligatory: xy is the variable name, it does not mean x*y.

Variable followed by ^ or ** is to be followed
by a positive integer and is a shortcut for the corresponding number of multiplications. E.g., x^3 and x**3 are equivalent to x*x*x.

Numeric coefficients are to start terms and are to be followed by the multiplication sign, e.g., x^2 - 2*x*y + y^2.

You can use spaces inside relations but not inside variable names or numeric coefficients.

- Max Degree

0 (zero) means that you do not restrict maximal degree to calculate Gröbner basis up to. A positive integer stops calculations
after receiving that degree. It is especially useful in the non-commutative mode where the basis is as the rule infinite.
A limitation of 100 is always applied in non-commutative case.

Your calculation will be also stopped if you exceed time or space limits. You can perform bigger calculations if you try
the background mode.

- Show results

Check boxes corresponding your desired results, and uncheck boxes for results you do not want to see.

The Gröbner basis is always calculated. You can select to show or not to show it.

In the commutative case, the Hilbert series is a rational function. You will see both representations, rational and series.

The Anick resolution can be calculated only in non-commutative mode.

- Background calculations

You can ask for backgroud calculations by checking the box '[ ] Send results by e-mail' and entering your e-mail address.
It may be desired due to poor communication speed, or too big volume of results, or time/space restrictions applied to
on-line demo calculations.

In that case, after clicking '[Go]'
you can return to the form and vary input data for the next calculation, or leave our page. Results will be sent you
by e-mail.

Of course you can download and install bergman on your own machine and
enjoy its full power.