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inverse image under ring homomorphism

R.<x,y> = QQ[]
R1 = R.quotient(R.ideal(y^2-x^3))
R2 = R.quotient(R.ideal(x))
h = R1.hom([y^2, y^3], R2)
h.inverse_image(y)
h.inverse_image(y^2)

Hi there! Just starting to learn sage and can't get why both queries above return "element y (y^2) does not have preimage" while y^2 clearly has x as preimage. Referring to the generators of R1 and R2 explicitly doesn't help either.

Thanks!

inverse image under ring homomorphism

R.<x,y> = QQ[]
R1 = R.quotient(R.ideal(y^2-x^3))
R2 = R.quotient(R.ideal(x))
h = R1.hom([y^2, y^3], R2)
h.inverse_image(y)
h.inverse_image(y^2)

Hi there! there!

Just starting to learn sage Sage and can't get why both queries above return "element y (y^2) inverse_image calls below give an error saying the given element does not have preimage" a preimage, while y^2 y^2 clearly has x x as preimage. preimage.

Referring to the generators of R1 R1 and R2 R2 explicitly doesn't help either.

Thanks!

sage: R.<x, y> = QQ[]
sage: R1 = R.quotient(R.ideal(y^2 - x^3))
sage: R2 = R.quotient(R.ideal(x))
sage: h = R1.hom([y^2, y^3], R2)
sage: h
Ring morphism:
  From: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (-x^3 + y^2)
  To:   Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x)
  Defn: xbar |--> y^2
        ybar |--> y^3

sage: h.inverse_image(y)
Traceback (most recent call last)
...
ValueError: element y does not have preimage

sage: h.inverse_image(y^2)
Traceback (most recent call last)
...
ValueError: element y^2 does not have preimage

Thanks!

inverse image under ring homomorphism

Hi there!

Just starting to learn Sage and can't get why both inverse_image calls below give an error saying the given element does not have a preimage, while y^2 clearly has xxbar as preimage.

Referring to the generators of R1 and R2 explicitly doesn't help either.

sage: R.<x, y> = QQ[]
sage: R1 = R.quotient(R.ideal(y^2 - x^3))
sage: R2 = R.quotient(R.ideal(x))
sage: h = R1.hom([y^2, y^3], R2)
sage: h
Ring morphism:
  From: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (-x^3 + y^2)
  To:   Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x)
  Defn: xbar |--> y^2
        ybar |--> y^3

sage: h.inverse_image(y)
Traceback (most recent call last)
...
ValueError: element y does not have preimage

sage: h.inverse_image(y^2)
Traceback (most recent call last)
...
ValueError: element y^2 does not have preimage

Thanks!

inverse image under ring homomorphism

Hi there!

Just starting to learn Sage and can't get why both inverse_image calls below give an error saying the given element does not have a preimage, while y^2 clearly has xbar as preimage.

Referring to the generators of R1 and R2 explicitly doesn't help either.

sage: R.<x, y> = QQ[]
sage: R1 = R.quotient(R.ideal(y^2 - x^3))
sage: R2 = R.quotient(R.ideal(x))
sage: h = R1.hom([y^2, y^3], R2)
sage: h
Ring morphism:
  From: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (-x^3 + y^2)
  To:   Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x)
  Defn: xbar |--> y^2
        ybar |--> y^3

sage: h.inverse_image(y)
Traceback (most recent call last)
...
ValueError: element y does not have preimage

sage: h.inverse_image(y^2)
Traceback (most recent call last)
...
ValueError: element y^2 does not have preimage

Thanks!