User:Akritas2/sandbox

Regarding the first matrix of 1840 I would like to add the following:

This matrix can be easily computed with the function $$sylvester(p, q, x, 1)$$ of the computer algebra system sympy. This function can be found in the sympy module https://github.com/sympy/sympy/blob/master/sympy/polys/subresultants_qq_zz.py. The optional argument 1 in the function denotes the Sylvester matrix of 1840. (The module can also be attach-ed in a sage session.)

Definition 2
A polynomial remainder sequence ($$prs$$) is complete if the degrees of any two consecutive polynomials differ by one.

The Sylvester matrix of 1840 is related with the polynomial remainder sequence ($$prs$$) obtained by the Euclidean algorithm for polynomials. The Euclidean $$prs$$ of two polynomials $$p, q\in Z[x]$$ can be computed in $$Z[x]$$ with the function $$euclid\_amv(p, q, x)$$ found also in the sympy module mentioned above.

If $$euclid\_amv(p, q, x)$$ is a complete Euclidean $$prs$$ then the coefficients of the polynomials in the sequence can be also computed as determinants of appropriately selected submatrices of $$sylvester(p, q, x, 1)$$.

Stated in other words, a complete Euclidean $$prs$$ is identical to the corresponding subresultants prs. That is, we have $$euclid\_amv(p, q, x) = subresultants\_amv(p, q, x)$$.

The above equality is not true for incomplete $$prs$$'s. The signs of the coefficients in $$euclid\_amv(p, q, x)$$ will, in general, differ from the corresponding ones in $$subresultants\_amv(p, q, x)$$.

Regarding the variant I would like to add the following:

This matrix can be easily computed with the function $$sylvester(p, q, x, 2)$$ of the sympy module mentioned above. The optional argument 2 in the function denotes the Sylvester matrix of 1853.

The Sylvester matrix of 1853 is related with the polynomial remainder sequence obtained by the Sturmian algorithm for polynomials. The Sturmian $$prs$$ of two polynomials $$p, q\in Z[x]$$ can be computed in $$Z[x]$$ with the function $$sturm\_amv(p, q, x)$$ of the sympy module mentioned above.

If $$sturm\_amv(p, q, x)$$ is a complete Sturmian $$prs$$ then the coefficients of the polynomials in the sequence can be also computed as determinants of appropriately selected submatrices of $$sylvester(p, q, x, 2)$$.

Stated in other words, a complete Sturmian $$prs$$ is identical to the corresponding sequence of modified subresultant prs. That is, we have $$sturm\_amv(p, q, x) = modified\_subresultants\_amv(p, q, x)$$.

The above equality is not true for incomplete $$prs$$'s. The signs of the coefficients in $$sturm\_amv(p, q, x)$$ will, in general, differ from the corresponding ones in $$modified\_subresultants\_amv(p, q, x)$$.