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> **Question 1.** Let $R$ be a polynomial ring over a field $k$. Assume that $R$ is graded in the usual way (i.e., each variable has degree $1$). Let $I$ and $J$ be two ideals of $R$ such that $I$ is generated in degree $\leq a$ (that is, there is a set of generators of $I$ that have degrees $\leq a$) and $J$ is generated in degree $\leq b$. Is it true that the ideal $I \cap J$ is generated in degree $\leq a+b$ ?
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When $R$ is a univariate polynomial ring, this is simply claiming that the lcm of two polynomials of degrees $\leq a$ and $\leq b$ is a polynomial of degree $\leq a+b$; of course this is correct. But this kind of logic does not generalize to more variables. (It is [easy to see](https://web.archive.org/web/20150512110802/https://crazyproject.wordpress.com/2011/02/27/the-intersection-of-two-monomial-ideals-is-a-monomial-ideal/) that Question 1 has a positive answer when $I$ and $J$ are monomial ideals.)


If true, Question 1 would give a new (inductive) proof for the following theorem (part of the Subspace arrangement theorem of Derksen and Sidman -- see Theorem 2.1 in [their *A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements*, arXiv:math/0109035v1](https://arxiv.org/abs/math/0109035v1)):



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> **Theorem 2.** Let $R$ be a polynomial ring over a field $k$. Assume that $R$ is graded in the usual way (i.e., each variable has degree $1$). Let $I\_1, I\_2, \ldots, I\_t$ be $t$ homogeneous ideals of $R$ that are all generated in degree $1$ (that is, each $I\_i$ has the form $I\_i = V\_i R$ for some $k$-vector subspace $V\_i$ of the degree-$1$ homogeneous component of $R$). Then, the ideal $I\_1 \cap I\_2 \cap \cdots \cap I\_t$ is generated in degree $\leq t$.
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Some more questions suggest themselves:



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> **Question 3.** (If the answer to Question 1 is negative:) Does the answer to Question 1 become positive if we replace "ideal" by "homogeneous ideal"?
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(This would still be enough to reprove Theorem 2.)


Truth be told, I'm less interested in the polynomial case than I am in the exterior algebra case:



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> **Question 4.** What if we replace the polynomial ring by an exterior algebra?
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(You can assume $k$ has characteristic $0$ for simplicity.)


Finally, if **all** these questions have negative answers, here is what I am really looking for:



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> **Question 5.** Is there an elementary proof of the analogue of Theorem 2 for exterior algebras (which is part of Theorem 9 in [Francesca Gandini, *Degree bounds for invariant skew polynomials*, arXiv:2108.01767v1](https://arxiv.org/abs/2108.01767v1))?
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"Elementary" means no use of nontrivial commutative algebra for me (I'm actually fine with Schur functors, although I have a hunch that they too can be avoided).


I'm putting the invariant theory tag on this question because the ultimate use of Question 5 is in Francesca Gandini's proof of the analogue of the Noether bound for the exterior algebra, which I have asked about in [Noether's bound for anticommutative invariant theory (diff. forms instead of polynomials)?](https://mathoverflow.net/questions/38565/) and am now trying to understand...



Intersecting a ideal generated in degree $\leq a$ with one generated in degree $\leq b$ in a polynomial ring
Take $I=(a^3,b^3)$ and $J=(ac^2-bd^2)$. Then according to Macaulay2, $I\cap J$ has generators in degrees $7,8,9$, for instance $a^3c^6-b^3d^6$. So the answers to Question 3 and 1 are no.


