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













3












$begingroup$


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







share|cite|improve this question











$endgroup$











  • $begingroup$
    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
    34 mins ago











  • $begingroup$
    Sure. Is that different from what I wrote?
    $endgroup$
    – abx
    31 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
    26 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
    9 mins ago
















3












$begingroup$


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







share|cite|improve this question











$endgroup$











  • $begingroup$
    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
    34 mins ago











  • $begingroup$
    Sure. Is that different from what I wrote?
    $endgroup$
    – abx
    31 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
    26 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
    9 mins ago














3












3








3





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 58 mins ago







abx

















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
    34 mins ago











  • $begingroup$
    Sure. Is that different from what I wrote?
    $endgroup$
    – abx
    31 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
    26 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
    9 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
    34 mins ago











  • $begingroup$
    Sure. Is that different from what I wrote?
    $endgroup$
    – abx
    31 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
    26 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
    9 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
34 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
34 mins ago













$begingroup$
Sure. Is that different from what I wrote?
$endgroup$
– abx
31 mins ago




$begingroup$
Sure. Is that different from what I wrote?
$endgroup$
– abx
31 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
26 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
26 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
9 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
9 mins ago











1 Answer
1






active

oldest

votes


















4












$begingroup$

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?






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thanks, but could you explain why the Koszul complex gives that?
    $endgroup$
    – abx
    27 mins ago










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









4












$begingroup$

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?






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thanks, but could you explain why the Koszul complex gives that?
    $endgroup$
    – abx
    27 mins ago















4












$begingroup$

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?






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Thanks, but could you explain why the Koszul complex gives that?
    $endgroup$
    – abx
    27 mins ago













4












4








4





$begingroup$

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?






share|cite|improve this answer









$endgroup$



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?







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 44 mins ago









SashaSasha

21k22755




21k22755











  • $begingroup$
    Thanks, but could you explain why the Koszul complex gives that?
    $endgroup$
    – abx
    27 mins ago
















  • $begingroup$
    Thanks, but could you explain why the Koszul complex gives that?
    $endgroup$
    – abx
    27 mins ago















$begingroup$
Thanks, but could you explain why the Koszul complex gives that?
$endgroup$
– abx
27 mins ago




$begingroup$
Thanks, but could you explain why the Koszul complex gives that?
$endgroup$
– abx
27 mins ago

















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