Removing multiple roots of multivariate polynomials
To remove multiple roots (i.e. roots with multiplicity greater than 1) of a univariate polynomial $p$, one divides $p$ by GCD($p$, $p'$).
Does there exists anything similar for bivariate polynomials? I read there is a multivariate GCD algorithm but what about the derivative? I only heard about univariate derivatives but I assume there is also a multivariate extension? (Are these the "partial derivatives"? If so, how does one make use of them?) Or does one proceed entirely differently?
My question is, how to remove multiple roots of a bivariate (or multivariate) polynomial? I would greatly appreciate a simple example such as:
$$p = x^2$$
$$p' = 2x$$
$$GCD(p, p') = x$$
$$p/GCD(p, p') = x$$
derivatives polynomials roots
asked Jan 4 at 17:40
Ecir HanaEcir Hana
406314
add a comment |
To remove multiple roots (i.e. roots with multiplicity greater than 1) of a univariate polynomial $p$, one divides $p$ by GCD($p$, $p'$).
Does there exists anything similar for bivariate polynomials? I read there is a multivariate GCD algorithm but what about the derivative? I only heard about univariate derivatives but I assume there is also a multivariate extension? (Are these the "partial derivatives"? If so, how does one make use of them?) Or does one proceed entirely differently?
My question is, how to remove multiple roots of a bivariate (or multivariate) polynomial? I would greatly appreciate a simple example such as:
$$p = x^2$$
$$p' = 2x$$
$$GCD(p, p') = x$$
$$p/GCD(p, p') = x$$
derivatives polynomials roots
asked Jan 4 at 17:40
Ecir HanaEcir Hana
406314
add a comment |
To remove multiple roots (i.e. roots with multiplicity greater than 1) of a univariate polynomial $p$, one divides $p$ by GCD($p$, $p'$).
Does there exists anything similar for bivariate polynomials? I read there is a multivariate GCD algorithm but what about the derivative? I only heard about univariate derivatives but I assume there is also a multivariate extension? (Are these the "partial derivatives"? If so, how does one make use of them?) Or does one proceed entirely differently?
My question is, how to remove multiple roots of a bivariate (or multivariate) polynomial? I would greatly appreciate a simple example such as:
$$p = x^2$$
$$p' = 2x$$
$$GCD(p, p') = x$$
$$p/GCD(p, p') = x$$
derivatives polynomials roots
asked Jan 4 at 17:40
Ecir HanaEcir Hana
406314
To remove multiple roots (i.e. roots with multiplicity greater than 1) of a univariate polynomial $p$, one divides $p$ by GCD($p$, $p'$).
Does there exists anything similar for bivariate polynomials? I read there is a multivariate GCD algorithm but what about the derivative? I only heard about univariate derivatives but I assume there is also a multivariate extension? (Are these the "partial derivatives"? If so, how does one make use of them?) Or does one proceed entirely differently?
My question is, how to remove multiple roots of a bivariate (or multivariate) polynomial? I would greatly appreciate a simple example such as:
$$p = x^2$$
$$p' = 2x$$
$$GCD(p, p') = x$$
$$p/GCD(p, p') = x$$
derivatives polynomials roots
derivatives polynomials roots
asked Jan 4 at 17:40
Ecir HanaEcir Hana
406314
asked Jan 4 at 17:40
Ecir HanaEcir Hana
406314
asked Jan 4 at 17:40
Ecir HanaEcir Hana
406314
asked Jan 4 at 17:40
Ecir HanaEcir Hana
406314
asked Jan 4 at 17:40
Ecir HanaEcir Hana
406314
406314
add a comment |
add a comment |
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