1 | initial version |

If under substitution you mean to entirely eliminate $y$, you can do so by computing resultant of your polynomial and $y^2-(x^3+x)$ with respect to $y$:

```
sage: R.<x,y> = PolynomialRing(QQ)
sage: (y^2+x*y+1).resultant(y^2-(x^3+x),y)
x^6 - x^5 + 2*x^4 + x^3 + x^2 + 2*x + 1
```

2 | No.2 Revision |

If under substitution you mean to entirely eliminate $y$, you can do so by computing resultant of your polynomial and $y^2-(x^3+x)$ with respect to $y$:

`sage: R.<x,y> = `~~PolynomialRing(QQ) ~~PolynomialRing(GF(43))
sage: (y^2+x*y+1).resultant(y^2-(x^3+x),y)
x^6 - x^5 + 2*x^4 + x^3 + x^2 + 2*x + 1

3 | No.3 Revision |

~~If under substitution you mean ~~It is also possible to entirely eliminate ~~$y$, you can do so ~~$y$ by computing resultant of your polynomial and $y^2-(x^3+x)$ with respect to $y$:

```
sage: R.<x,y> = PolynomialRing(GF(43))
sage: (y^2+x*y+1).resultant(y^2-(x^3+x),y)
x^6 - x^5 + 2*x^4 + x^3 + x^2 + 2*x + 1
```

4 | No.4 Revision |

~~It ~~If your polynomial is assumed to be zero, then it is also possible to entirely eliminate $y$ by computing resultant of your polynomial and $y^2-(x^3+x)$ with respect to $y$:

```
sage: R.<x,y> = PolynomialRing(GF(43))
sage: (y^2+x*y+1).resultant(y^2-(x^3+x),y)
x^6 - x^5 + 2*x^4 + x^3 + x^2 + 2*x + 1
```

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