Relations between homogeneous polynomialsCount the number of homogeneous polynomialshomogeneous polynomials over a finite fieldIs complete homogeneous symmetric polynomials, an irreducibile element in Polynomial ring?Hilbert's Nullstellensatz on polynomials with integer coefficientsIs an ideal generated by multilinear, irreducible, homogeneous polynomials of different degrees always radical?morphism flat and relationsDegrees of generators of radical idealsDegrees of polynomials defining a Jacobian of maximal rank on a varietyReduction formula for Schubert polynomialsMinimal “subset” of a set of homogeneous polynomials with same solution space
Relations between homogeneous polynomials
Count the number of homogeneous polynomialshomogeneous polynomials over a finite fieldIs complete homogeneous symmetric polynomials, an irreducibile element in Polynomial ring?Hilbert's Nullstellensatz on polynomials with integer coefficientsIs an ideal generated by multilinear, irreducible, homogeneous polynomials of different degrees always radical?morphism flat and relationsDegrees of generators of radical idealsDegrees of polynomials defining a Jacobian of maximal rank on a varietyReduction formula for Schubert polynomialsMinimal “subset” of a set of homogeneous polynomials with same solution space
$begingroup$
Let $F_1,ldots ,F_ellin mathbbC[X_1,ldots , X_n]$ be general homogeneous polynomials of degree $d$. For $p<d$, it seems likely that there is no nontrivial relation $sum F_iG_i=0$ with $G_1,ldots ,G_ell$ homogeneous of degree $p$. Is this known? If yes, what would be a reference (or a proof)?
ag.algebraic-geometry ac.commutative-algebra
asked 1 hour ago
abxabx
23.8k34885
$endgroup$
add a comment |
$begingroup$
Let $F_1,ldots ,F_ellin mathbbC[X_1,ldots , X_n]$ be general homogeneous polynomials of degree $d$. For $p<d$, it seems likely that there is no nontrivial relation $sum F_iG_i=0$ with $G_1,ldots ,G_ell$ homogeneous of degree $p$. Is this known? If yes, what would be a reference (or a proof)?
ag.algebraic-geometry ac.commutative-algebra
asked 1 hour ago
abxabx
23.8k34885
$endgroup$
$begingroup$
I think this question needs some clarification. I think you may mean something along the lines of: let $X$ be the moduli space of $n$-tuples of homogeneous polynomials of degree $d$; within $X$, there is a subspace $Y$ consisting of homogeneous polynomials that satisfy a nontrivial relation $sum F_i G_i = 0$ for some $G_i$ homogeneous of degree $p < d$ (i.e. the union of the subspaces determined by each tuple $G$). Is it true that $Y subsetneq X$?
$endgroup$
– user44191
49 mins ago
$begingroup$
Sure. Is that different from what I wrote?
$endgroup$
– abx
46 mins ago
$begingroup$
I found what you wrote ambiguous or underdetermined, e.g. with what you mean by "general"; I thought what I wrote made it more explicitly clear (though it may be overkill as written).
$endgroup$
– user44191
42 mins ago
$begingroup$
This seem trivially false if I haven't misunderstood the question. Take $n = 1, p = 0, d = 1, ell = 2$. Then $F_1 = ax, F_2 = bx$ for some $a, b$, so $b F_1 - a F_2 = 0$. I think both $n, ell$ play some role.
$endgroup$
– user44191
25 mins ago
$begingroup$
Oh, right. $ellleq n$ would be fine for me, though this can be certainly weakened.
$endgroup$
– abx
3 mins ago
add a comment |
$begingroup$
Let $F_1,ldots ,F_ellin mathbbC[X_1,ldots , X_n]$ be general homogeneous polynomials of degree $d$. For $p<d$, it seems likely that there is no nontrivial relation $sum F_iG_i=0$ with $G_1,ldots ,G_ell$ homogeneous of degree $p$. Is this known? If yes, what would be a reference (or a proof)?
ag.algebraic-geometry ac.commutative-algebra
asked 1 hour ago
abxabx
23.8k34885
$endgroup$
Let $F_1,ldots ,F_ellin mathbbC[X_1,ldots , X_n]$ be general homogeneous polynomials of degree $d$. For $p<d$, it seems likely that there is no nontrivial relation $sum F_iG_i=0$ with $G_1,ldots ,G_ell$ homogeneous of degree $p$. Is this known? If yes, what would be a reference (or a proof)?
ag.algebraic-geometry ac.commutative-algebra
ag.algebraic-geometry ac.commutative-algebra
asked 1 hour ago
abxabx
23.8k34885
asked 1 hour ago
abxabx
23.8k34885
edited 1 hour ago
abx
asked 1 hour ago
abxabx
23.8k34885
asked 1 hour ago
abxabx
23.8k34885
asked 1 hour ago
abxabx
23.8k34885
23.8k34885
$begingroup$
I think this question needs some clarification. I think you may mean something along the lines of: let $X$ be the moduli space of $n$-tuples of homogeneous polynomials of degree $d$; within $X$, there is a subspace $Y$ consisting of homogeneous polynomials that satisfy a nontrivial relation $sum F_i G_i = 0$ for some $G_i$ homogeneous of degree $p < d$ (i.e. the union of the subspaces determined by each tuple $G$). Is it true that $Y subsetneq X$?
$endgroup$
– user44191
49 mins ago
$begingroup$
Sure. Is that different from what I wrote?
$endgroup$
– abx
46 mins ago
$begingroup$
I found what you wrote ambiguous or underdetermined, e.g. with what you mean by "general"; I thought what I wrote made it more explicitly clear (though it may be overkill as written).
$endgroup$
– user44191
42 mins ago
$begingroup$
This seem trivially false if I haven't misunderstood the question. Take $n = 1, p = 0, d = 1, ell = 2$. Then $F_1 = ax, F_2 = bx$ for some $a, b$, so $b F_1 - a F_2 = 0$. I think both $n, ell$ play some role.
$endgroup$
– user44191
25 mins ago
$begingroup$
Oh, right. $ellleq n$ would be fine for me, though this can be certainly weakened.
$endgroup$
– abx
3 mins ago
add a comment |
$begingroup$
I think this question needs some clarification. I think you may mean something along the lines of: let $X$ be the moduli space of $n$-tuples of homogeneous polynomials of degree $d$; within $X$, there is a subspace $Y$ consisting of homogeneous polynomials that satisfy a nontrivial relation $sum F_i G_i = 0$ for some $G_i$ homogeneous of degree $p < d$ (i.e. the union of the subspaces determined by each tuple $G$). Is it true that $Y subsetneq X$?
$endgroup$
– user44191
49 mins ago
$begingroup$
Sure. Is that different from what I wrote?
$endgroup$
– abx
46 mins ago
$begingroup$
I found what you wrote ambiguous or underdetermined, e.g. with what you mean by "general"; I thought what I wrote made it more explicitly clear (though it may be overkill as written).
$endgroup$
– user44191
42 mins ago
$begingroup$
This seem trivially false if I haven't misunderstood the question. Take $n = 1, p = 0, d = 1, ell = 2$. Then $F_1 = ax, F_2 = bx$ for some $a, b$, so $b F_1 - a F_2 = 0$. I think both $n, ell$ play some role.
$endgroup$
– user44191
25 mins ago
$begingroup$
Oh, right. $ellleq n$ would be fine for me, though this can be certainly weakened.
$endgroup$
– abx
3 mins ago
$begingroup$
I think this question needs some clarification. I think you may mean something along the lines of: let $X$ be the moduli space of $n$-tuples of homogeneous polynomials of degree $d$; within $X$, there is a subspace $Y$ consisting of homogeneous polynomials that satisfy a nontrivial relation $sum F_i G_i = 0$ for some $G_i$ homogeneous of degree $p < d$ (i.e. the union of the subspaces determined by each tuple $G$). Is it true that $Y subsetneq X$?
$endgroup$
– user44191
49 mins ago
$begingroup$
I think this question needs some clarification. I think you may mean something along the lines of: let $X$ be the moduli space of $n$-tuples of homogeneous polynomials of degree $d$; within $X$, there is a subspace $Y$ consisting of homogeneous polynomials that satisfy a nontrivial relation $sum F_i G_i = 0$ for some $G_i$ homogeneous of degree $p < d$ (i.e. the union of the subspaces determined by each tuple $G$). Is it true that $Y subsetneq X$?
$endgroup$
– user44191
49 mins ago
$begingroup$
Sure. Is that different from what I wrote?
$endgroup$
– abx
46 mins ago
$begingroup$
Sure. Is that different from what I wrote?
$endgroup$
– abx
46 mins ago
$begingroup$
I found what you wrote ambiguous or underdetermined, e.g. with what you mean by "general"; I thought what I wrote made it more explicitly clear (though it may be overkill as written).
$endgroup$
– user44191
42 mins ago
$begingroup$
I found what you wrote ambiguous or underdetermined, e.g. with what you mean by "general"; I thought what I wrote made it more explicitly clear (though it may be overkill as written).
$endgroup$
– user44191
42 mins ago
$begingroup$
This seem trivially false if I haven't misunderstood the question. Take $n = 1, p = 0, d = 1, ell = 2$. Then $F_1 = ax, F_2 = bx$ for some $a, b$, so $b F_1 - a F_2 = 0$. I think both $n, ell$ play some role.
$endgroup$
– user44191
25 mins ago
$begingroup$
This seem trivially false if I haven't misunderstood the question. Take $n = 1, p = 0, d = 1, ell = 2$. Then $F_1 = ax, F_2 = bx$ for some $a, b$, so $b F_1 - a F_2 = 0$. I think both $n, ell$ play some role.
$endgroup$
– user44191
25 mins ago
$begingroup$
Oh, right. $ellleq n$ would be fine for me, though this can be certainly weakened.
$endgroup$
– abx
3 mins ago
$begingroup$
Oh, right. $ellleq n$ would be fine for me, though this can be certainly weakened.
$endgroup$
– abx
3 mins ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I think this can be controlled as follows. Let $Z subset mathbbP^n-1$ be the complete intersection defined by $F_i$. Then there is a Koszul resolution
$$
dots to mathcalO(-2d)^binomell2 to mathcalO(-d)^ell to mathcalO to mathcalO_Z to 0.
$$
Twisting it by $mathcalO(d+p)$ we obtain
$$
dots to mathcalO(p-d)^binomell2 to mathcalO(p)^ell to mathcalO(d+p) to mathcalO_Z(d+p) to 0.
$$
Your question is equivalent to injectivity of the induced map
$$
H^0(mathcalO(p)^ell) to H^0(mathcalO(d+p)).
$$
If $n ge ell$ the cohomology spectral sequence of the twisted Koszul complex proves this. Is that enough for your purposes?
answered 1 hour ago
SashaSasha
21k22755
$endgroup$
1
$begingroup$
Thanks, but could you explain why the Koszul complex gives that?
$endgroup$
– abx
43 mins ago
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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votes
$begingroup$
I think this can be controlled as follows. Let $Z subset mathbbP^n-1$ be the complete intersection defined by $F_i$. Then there is a Koszul resolution
$$
dots to mathcalO(-2d)^binomell2 to mathcalO(-d)^ell to mathcalO to mathcalO_Z to 0.
$$
Twisting it by $mathcalO(d+p)$ we obtain
$$
dots to mathcalO(p-d)^binomell2 to mathcalO(p)^ell to mathcalO(d+p) to mathcalO_Z(d+p) to 0.
$$
Your question is equivalent to injectivity of the induced map
$$
H^0(mathcalO(p)^ell) to H^0(mathcalO(d+p)).
$$
If $n ge ell$ the cohomology spectral sequence of the twisted Koszul complex proves this. Is that enough for your purposes?
answered 1 hour ago
SashaSasha
21k22755
$endgroup$
1
$begingroup$
Thanks, but could you explain why the Koszul complex gives that?
$endgroup$
– abx
43 mins ago
add a comment |
$begingroup$
I think this can be controlled as follows. Let $Z subset mathbbP^n-1$ be the complete intersection defined by $F_i$. Then there is a Koszul resolution
$$
dots to mathcalO(-2d)^binomell2 to mathcalO(-d)^ell to mathcalO to mathcalO_Z to 0.
$$
Twisting it by $mathcalO(d+p)$ we obtain
$$
dots to mathcalO(p-d)^binomell2 to mathcalO(p)^ell to mathcalO(d+p) to mathcalO_Z(d+p) to 0.
$$
Your question is equivalent to injectivity of the induced map
$$
H^0(mathcalO(p)^ell) to H^0(mathcalO(d+p)).
$$
If $n ge ell$ the cohomology spectral sequence of the twisted Koszul complex proves this. Is that enough for your purposes?
answered 1 hour ago
SashaSasha
21k22755
$endgroup$
1
$begingroup$
Thanks, but could you explain why the Koszul complex gives that?
$endgroup$
– abx
43 mins ago
add a comment |
$begingroup$
I think this can be controlled as follows. Let $Z subset mathbbP^n-1$ be the complete intersection defined by $F_i$. Then there is a Koszul resolution
$$
dots to mathcalO(-2d)^binomell2 to mathcalO(-d)^ell to mathcalO to mathcalO_Z to 0.
$$
Twisting it by $mathcalO(d+p)$ we obtain
$$
dots to mathcalO(p-d)^binomell2 to mathcalO(p)^ell to mathcalO(d+p) to mathcalO_Z(d+p) to 0.
$$
Your question is equivalent to injectivity of the induced map
$$
H^0(mathcalO(p)^ell) to H^0(mathcalO(d+p)).
$$
If $n ge ell$ the cohomology spectral sequence of the twisted Koszul complex proves this. Is that enough for your purposes?
answered 1 hour ago
SashaSasha
21k22755
$endgroup$
I think this can be controlled as follows. Let $Z subset mathbbP^n-1$ be the complete intersection defined by $F_i$. Then there is a Koszul resolution
$$
dots to mathcalO(-2d)^binomell2 to mathcalO(-d)^ell to mathcalO to mathcalO_Z to 0.
$$
Twisting it by $mathcalO(d+p)$ we obtain
$$
dots to mathcalO(p-d)^binomell2 to mathcalO(p)^ell to mathcalO(d+p) to mathcalO_Z(d+p) to 0.
$$
Your question is equivalent to injectivity of the induced map
$$
H^0(mathcalO(p)^ell) to H^0(mathcalO(d+p)).
$$
If $n ge ell$ the cohomology spectral sequence of the twisted Koszul complex proves this. Is that enough for your purposes?
answered 1 hour ago
SashaSasha
21k22755
answered 1 hour ago
SashaSasha
21k22755
answered 1 hour ago
SashaSasha
21k22755
answered 1 hour ago
SashaSasha
21k22755
21k22755
1
$begingroup$
Thanks, but could you explain why the Koszul complex gives that?
$endgroup$
– abx
43 mins ago
add a comment |
1
$begingroup$
Thanks, but could you explain why the Koszul complex gives that?
$endgroup$
– abx
43 mins ago
1
1
$begingroup$
Thanks, but could you explain why the Koszul complex gives that?
$endgroup$
– abx
43 mins ago
$begingroup$
Thanks, but could you explain why the Koszul complex gives that?
$endgroup$
– abx
43 mins ago
add a comment |
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$begingroup$
I think this question needs some clarification. I think you may mean something along the lines of: let $X$ be the moduli space of $n$-tuples of homogeneous polynomials of degree $d$; within $X$, there is a subspace $Y$ consisting of homogeneous polynomials that satisfy a nontrivial relation $sum F_i G_i = 0$ for some $G_i$ homogeneous of degree $p < d$ (i.e. the union of the subspaces determined by each tuple $G$). Is it true that $Y subsetneq X$?
$endgroup$
– user44191
49 mins ago
$begingroup$
Sure. Is that different from what I wrote?
$endgroup$
– abx
46 mins ago
$begingroup$
I found what you wrote ambiguous or underdetermined, e.g. with what you mean by "general"; I thought what I wrote made it more explicitly clear (though it may be overkill as written).
$endgroup$
– user44191
42 mins ago
$begingroup$
This seem trivially false if I haven't misunderstood the question. Take $n = 1, p = 0, d = 1, ell = 2$. Then $F_1 = ax, F_2 = bx$ for some $a, b$, so $b F_1 - a F_2 = 0$. I think both $n, ell$ play some role.
$endgroup$
– user44191
25 mins ago
$begingroup$
Oh, right. $ellleq n$ would be fine for me, though this can be certainly weakened.
$endgroup$
– abx
3 mins ago