<Text-field style="Heading 1" layout="Heading 1">Introduction to Groebner Bases</Text-field>

<Text-field style="Heading 1" layout="Heading 1">Polynomial Long Division</Text-field>

Back to question 3. How do we test if there are any redundant equations in these systems. We would like to remove the redundant equations to simplify the system.

sys1;

PCYvKiZJInhHNiIiIiJJInlHRiZGJ0kiekdGJi8sJiokKUYlIiIjRidGJyokKUYoRi5GJ0YnRicvLCZGLCEiIiokKUYpRi5GJ0YnRicvLCZGL0YnRjRGJ0Yu

sys3;

PCcvLCZJIndHNiIiIiJJInhHRiZGJyIiIS8sJkYlRidJInpHRiYhIiJGJy8sJkkieUdGJkYnRixGJ0YnLywoRihGJ0YwRicqJiIiI0YnRixGJ0YnRikvLCpGJUYnRihGJ0YwRidGLEYnRic=

Let's recall division. Shown in the figure below (a Maple canvas, very useful for creating a figure) is a division of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5 divided LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= using the highschool long division algorithm as I remember seeing it.

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We obtain the quotient LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ4RidGL0YyLUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GVi1JI21uR0YkNiRRIjFGJ0Y5Rjk= and remainder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk= satisfying 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 . Have you every wondered why we always work with the terms sorted in decreasing degree? If you try executing the division algorithm with the terms sorted in increasing degree, you'll find that it works if the remainder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= but doesn't terminate if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=. To make the division algorithm work for polynomials in more than one variable we will need to sort the terms. But we will also need to be able to divide by more than one divisor. You may already have seen this in linear algebra. Consider the four vectors

u1,u2,u3,u4 := <1,1,0,0>, <1,0,0,-1>, <0,0,1,1>, <0,1,1,2>;

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

Suppose we are asked if there is a linear dependency between them? My son Philip is taking his first course in Linear Algebra at the University of British Columbia this Fall. He asked me how to do this last week! I had to try to remember. There is more than one way to do it. One way is to first see if 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 are linearly independent. I'll do this using row Echelon form. First I put the vectors 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 as rows of a matrix then reduce it to Row Echelon form.

A := Matrix( <Transpose(u1), Transpose(u2), Transpose(u3)> );

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

R := ReducedRowEchelonForm( A );

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

This matrix has rank 3 so yes, 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 are linearly independent. Now what about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiNEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+? Is it also linearly independent? One way is to set up a linear system from 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 and solve for the unknowns LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiYkYnRi9GMkY1LUYsNiVRImNGJ0YvRjJGOQ== . A second way (better) is to use the matrix R and divide the vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiNEYnL0Y2USdub3JtYWxGJ0YyRjUtRiM2JS1GLzYlUSJURidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGRy9JK21zZW1hbnRpY3NHRiRRLFtub25lLG5vbmVdRidGPg== by the rows of R? How? By subtracting multiples of the rows of R from 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 reduce 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 as follows.

Transpose(u4);

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

Transpose(u4) - R[2];

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

Transpose(u4) - R[2] - R[3];

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

The remainder is 0 therefore there is a dependency. We are doing a division. Repeated subtraction is division. If we had done this with the polynomials instead of with row vectors then it would look more like a polynomial division. Let's turn to the general division algorithm.

<Text-field style="Heading 1" layout="Heading 1">The General Division Algorithm</Text-field>

It may surprise you to learn that this division algorithm was not known before the 20th century. It is simple but it does depend crucially on how we order the terms in our polynomials. Anyone can learn how to do this. When we were dividing vectors, we used an implicit ordering on the unknowns LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRi9GMkY1LUYsNiVRInlGJ0YvRjJGNS1GLDYlUSJ6RidGL0YyRjk= , namely LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ4RidGL0YyRjUtRiw2JVEieUYnRi9GMkY1LUYsNiVRInpGJ0YvRjJGOQ== because of the order we put on the columns of A. The first step to generalize the division algorithm to polynomials in more than one variable is to decide how to order terms. Suppose the terms of the polynomial are

f := x^2,y^2,4*x*y^2,5*x,2,7*x*y;

NigqJClJInhHNiIiIiMiIiIqJClJInlHRiZGJ0YoLCQqKCIiJUYoRiVGKEYqRihGKCwkKiYiIiZGKEYlRihGKEYnLCQqKCIiKEYoRiVGKEYrRihGKA==

One way to order the terms is to first sort on the degree in the first variable, say x. So we have (in descending degree)

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

We have one term of degree 2 in x, three of degree 1 and two of degree 0. Now for each set of terms with the same degree in x, sort them in decreasing degree in y to get

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

This ordering is called pure lexicographical order. In Maple we specify this using LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVElcGxleEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGQUZBRkE=. If there were a third variable z we would now sort all terms in 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 in descending degree on z. The first term, the leading term, is key to the division algorithm. We will denote the leading term of a polynomial f by LT(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=). Now when I input the polynomial in this ordering to Maple I get

f := x^2+4*x*y^2+7*x*y+5*x+y^2+2;

LC4qKCIiJSIiIkkieEc2IkYlKUkieUdGJyIiI0YlRiUqJClGJkYqRiVGJSooIiIoRiVGJkYlRilGJUYlKiRGKEYlRiUqJiIiJkYlRiZGJUYlRipGJQ==

Notice that it changed the order of the terms and put the term 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 first. Why? Because it is using a different term ordering that will also work for the division algorithm and for Groebner bases. Consider again our terms

f := [x^2,y^2,4*x*y^2,5*x,2,7*x*y];

NygqJClJInhHNiIiIiMiIiIqJClJInlHRiZGJ0YoLCQqKCIiJUYoRiVGKEYqRihGKCwkKiYiIiZGKEYlRihGKEYnLCQqKCIiKEYoRiVGKEYrRihGKA==

This time, first sort the terms by degree (into descending order). So the term 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 has degree 3 which is more than the term LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1JJW1zdXBHRiQ2JS1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiM2JS1GLDYkUSIyRidGL0ZRRlQvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRi8= which has degree 2. We get

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

There is 1 term of degree 3, three of degree 2, one of degree 1 and one of degree 0. Now sort the terms of each degree by lexicographical order to get

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.

This term ordering is called graded lexicographical order. To get a Groebner basis in this ordering use grlex(x,y). Just to see that this makes a difference let's compare the two orderings

Groebner[Basis]( B1, plex(x,y,z) );

NyUsKCokKUkiekc2IiIiJSIiIkYpKiYiIiNGKSlGJkYrRikhIiJGK0YpLCgqJClJInlHRidGK0YpRikqJEYsRilGKUYrRi0sJiomRjFGKSlGJiIiJEYpRi0qJkYrRilJInhHRidGKUYp

Groebner[Basis]( B1, grlex(x,y,z) );

NygsKCokKUkieUc2IiIiIyIiIkYpKiQpSSJ6R0YnRihGKUYpRighIiIsJiomSSJ4R0YnRilGJkYpRilGLEYtLCgqJClGMEYoRilGKUYqRi1GKUYpLCgqJkYmRilGK0YpRikqJkYwRilGLEYpRi1GJkYtLCgqJkYwRilGK0YpRikqJkYmRilGLEYpRikqJkYoRilGMEYpRi0sKCokKUYsIiIlRilGKSomRihGKUYrRilGLUYoRik=

Now we are ready to do a general division. We will divide the polynomial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RidGL0YyLUYjNiUtSSNtbkdGJDYkUSIyRidGOUYvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GXG9GUi1GNjYtUScmc2RvdDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GYm8tRlA2JS1GLDYlUSJ5RidGL0YyRlVGZW5GaG4tRlA2JS1GLDYlUSJ6RidGL0YyRlVGZW4tRjY2LVEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSEZbb0Zdby1GWDYkUSIxRidGOUY5 by two polynomials 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 . The first step is to sort the terms so we can identify the leading terms of the dividend LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and divisors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIUYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGWEY+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ . We have

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 divided by {LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8=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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RKCZtaW51cztGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlgtRi82JVEieEYnRjJGNUZDLUYvNiVRInpGJ0YyRjVGPg== }.

The leading terms are LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= and for the divisors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= . The division algorithm always works with leading terms. It will produce two quotients LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and one remainder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= which will satisfy

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

and either LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= or no term in the remainder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is divisable by any leading term of a divisor.

The following figure shows the first step of the division algorithm. I've put the two divisors on the left and the two quotients on the top of the radical symbol and the dividend LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the usual place. The first quotient term is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlhGOkY+ . We multiply and subtract 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 as in the division algorithm for one variable.

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Fdp-Ffu6'Fhu-FA6#$"#IF--FF6&FH-FQ6#-FT6$$"%XHF-$"$0*F-FZFinFiv-Fco6$7$F\r$"$+*F-7$$"%!*QF-$"$5*F--Ffu6'FhuFey-FF6&FH-FQ6#-FT6$$"%S>F-$"%57F-FZFinFiv-Fco6$7$F\rF[[l7$$"%!)=F-$"%5:F-

Notice that after the subtraction we sort the terms in the result so that we can identify the leading term of the result which is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJyZzZG90O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUklbXN1cEdGJDYlLUYsNiVRInlGJ0YvRjItRiM2JS1JI21uR0YkNiRRIjJGJ0Y5Ri9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGOQ== . 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 does not divide LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJyZzZG90O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUklbXN1cEdGJDYlLUYsNiVRInlGJ0YvRjItRiM2JS1JI21uR0YkNiRRIjJGJ0Y5Ri9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGOQ== but 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 does with quotient LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. So we add the next quotient term LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and multiply and subtract. Here are the next two steps

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Fdp

At this point we have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRjs2JFEiMkYnRj5GPg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRi82JVEieEYnRjJGNUY+ and no terms 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 are divisible by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= the leading terms of the divisors so this is the remainder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= .

Let us do another division. Consider

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 divided by { 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 and 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 }

using lexicographical order with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ5RidGL0YyRjk= so that the terms are already sorted. Since 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 and 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 both divide 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 so the result is going to depend on which we use first. Let's try it both ways to see what happens.

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Note, the terms in the remainder 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 are not divisible by either 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 or 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.

What is surprising is that in one order we get 0 for the remainder but in the other we don't. And that is a serious problem. It means that there's something wrong with the division algorithm. We can't use it as expected to test if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNiYtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYkUSIxRicvRjtRJ25vcm1hbEYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiLEYnRkMvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGTi8lKnN5bW1ldHJpY0dGTi8lKGxhcmdlb3BHRk4vJS5tb3ZhYmxlbGltaXRzR0ZOLyUnYWNjZW50R0ZOLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JUYzLUYjNiUtRkA2JFEiMkYnRkNGN0Y6RkVGQ0ZDLyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJy1GSTYtUSJ+RidGQ0ZML0ZQRk5GUUZTRlVGV0ZZRmVuL0ZpbkZnbkZD divides LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= with 0 remainder. It turns out that the problem is not the division algorithm, rather, it's the basis LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNiYtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYkUSIxRicvRjtRJ25vcm1hbEYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiLEYnRkMvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGTi8lKnN5bW1ldHJpY0dGTi8lKGxhcmdlb3BHRk4vJS5tb3ZhYmxlbGltaXRzR0ZOLyUnYWNjZW50R0ZOLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JUYzLUYjNiUtRkA2JFEiMkYnRkNGN0Y6RkVGQ0ZDLyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJy1GSTYtUSFGJ0ZDRkwvRlBGTkZRRlNGVUZXRllGZW4vRmluRmduRkM= that we are using. It's not a "good basis". That is, it's "not a Groebner basis". If we'd used a Groebner basis we would get a 0 remainder. Incidentally, the same problem occurs if you try to divide a vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= by a set of vectors 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 if you don't first put the vectors in Row Echelon form. Let us first simplify the divisor basis. We have

g1 := x*y+y;

LCYqJkkieEc2IiIiIkkieUdGJUYmRiZGJ0Ym

g2 := x*y+x;

LCYqJkkieEc2IiIiIkkieUdGJUYmRiZGJEYm

g3 := g2-g1;

LCZJInhHNiIiIiJJInlHRiQhIiI=

I'd better use g3 to simplify g1 and g2

g1 := g1 - expand(y*g3); # simplify g1

LCYqJClJInlHNiIiIiMiIiJGKEYlRig=

g2 := g2 - expand(y*g3); # simplify g2

LCYqJClJInlHNiIiIiMiIiJGKEkieEdGJkYo

g2 := g2-g3; # simnplify g2 some more

LCYqJClJInlHNiIiIiMiIiJGKEYlRig=

G := {g1,g2,g3};

PCQsJkkieEc2IiIiIkkieUdGJSEiIiwmKiQpRiciIiNGJkYmRidGJg==

This is actually a Groebner basis and we have the following property.

Property of a Groebner Basis. A Groebner basis 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 for a set of input polynomials 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 with respect to a given term ordering, has the property that if a given polynomial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= may be written as 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 for some polynomials LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn , then the remainder of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= divided by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is 0. This allows us to answer question 3.

sys1;

PCYvKiZJInhHNiIiIiJJInlHRiZGJ0kiekdGJi8sJiokKUYlIiIjRidGJyokKUYoRi5GJ0YnRicvLCZGLCEiIiokKUYpRi5GJ0YnRicvLCZGL0YnRjRGJ0Yu

f1,f2,f3,f4 := x*y-z,x^2+y^2-1,z^2-x^2-1,y^2+z^2-2;

NiYsJiomSSJ4RzYiIiIiSSJ5R0YmRidGJ0kiekdGJiEiIiwoKiQpRiUiIiNGJ0YnKiQpRihGLkYnRidGJ0YqLChGLEYqKiQpRilGLkYnRidGJ0YqLChGL0YnRjJGJ0YuRio=

To test if the 4th equation is redundant, we first compute a Groebner basis G for 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 using any term ordering.

G := Groebner[Basis]( [f1,f2,f3], plex(x,y,z) );

NyUsKCokKUkiekc2IiIiJSIiIkYpKiYiIiNGKSlGJkYrRikhIiJGK0YpLCgqJClJInlHRidGK0YpRikqJEYsRilGKUYrRi0sJiomRjFGKSlGJiIiJEYpRi0qJkYrRilJInhHRidGKUYp

Now we just need to divide LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiNEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and see if we get 0 remainder. This command does the division and returns the remainder

Groebner[NormalForm]( f4, G, plex(x,y,z) );

IiIh

Since it is 0 then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiNEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ is redundant.

Note, division of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiNEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ by 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 does not yield a 0 remainder

Groebner[NormalForm]( f4, [f1,f2,f3], plex(x,y,z) );

LCgqJClJInlHNiIiIiMiIiJGKCokKUkiekdGJkYnRihGKEYnISIi