Advanced Groebner Basis Calculator

Calculator Form

Enter polynomials without parentheses. Use nonnegative powers only. You may write x*y-1 or xy-1. This calculator supports up to three variables and six input polynomials.

Variables

Monomial Order

Max Pair Iterations

Display Precision

Zero Tolerance

Show detailed reduction steps

Polynomial 1

Polynomial 2

Polynomial 3

Polynomial 4

Polynomial 5

Polynomial 6

Example Data Table

Variables	Order	Input Polynomial 1	Input Polynomial 2	Reduced Basis Output
x, y	Lexicographic	x*y-1	y^2-x	x-y^2 ; y^3-1

This sample demonstrates elimination behavior under lexicographic order. The final basis contains one polynomial solving for x directly and another using only y.

Formula Used

Leading term: LT(f) = LC(f) · LM(f)

S-polynomial: S(f,g) = (LCM(LM(f),LM(g))/LM(f))·f/LC(f) - (LCM(LM(f),LM(g))/LM(g))·g/LC(g)

Reduction: repeatedly cancel the current leading term using basis elements whose leading monomials divide it.

The calculator applies a practical Buchberger workflow. It parses the entered polynomial system, normalizes nonzero inputs, creates S-polynomials for active pairs, reduces each remainder, and adds any nonzero remainder into the basis.

After pair processing finishes, the calculator reduces each basis element by the others and rescales every survivor to a monic form. That final set is reported as the reduced Groebner basis.

How to Use This Calculator

Enter one to three variables, separated by commas.
Select a monomial order that matches your goal.
Type up to six polynomials using the chosen variables.
Use * optionally and powers like x^3.
Set the pair limit, precision, and tolerance.
Submit the form to compute the reduced basis.
Review the resulting basis, basis table, and graph.
Download the basis summary through CSV or PDF buttons.

Frequently Asked Questions

1) What does this calculator compute?

It computes a practical reduced Groebner basis for the entered polynomial ideal, along with leading terms, term counts, degrees, step logs, and exportable summaries.

2) Which monomial order should I choose?

Use lexicographic order for elimination-style results, graded lexicographic for degree-first comparisons, and graded reverse lexicographic when you want a common computational choice.

3) Why can the computation stop early?

Large systems may create many S-polynomials. The pair limit prevents very slow runs on a web page. Increase the limit when you need deeper computation.

4) Why do some coefficients appear as decimals?

The calculator uses floating-point arithmetic for browser-friendly performance. Precision settings control display rounding, while tolerance removes tiny numerical leftovers near zero.

5) Can I use this to solve polynomial systems?

Yes. A Groebner basis often rewrites a system into easier elimination forms. You can inspect single-variable equations, back-substitute, and study ideal structure.

6) What is a reduced Groebner basis?

It is a normalized basis where every leading coefficient is one and no term of one basis polynomial is reducible by another basis polynomial.

7) Why might two bases look different?

Changing the variable order or monomial order changes leading terms. Different leading terms produce different S-polynomials and therefore different Groebner bases.

8) What does the graph show?

The graph compares basis complexity. Bars show how many terms each basis polynomial contains, and the line shows the total degree.