Hun glemmer sine piller på bordet, så hun må hente dem igen.

Sine is the reflexive possessive in Danish: it refers back to the subject of the same clause.

• Hun glemmer sine piller = she forgets her own pills.
  Hendes is non-reflexive and typically points to someone else’s possession (or can be ambiguous without context):
• Hun glemmer hendes piller would usually be understood as “she forgets another woman’s pills.”

They all mean “his/her/its/their own,” and you choose based on the gender/number of the noun possessed:

• sin
  • common gender singular noun (en-words): sin bil
• sit
  • neuter singular noun (et-words): sit hus
• sine
  • plural noun: sine piller (because piller is plural)

Here så links two independent clauses (each has its own subject + verb):

• Hun glemmer ...,
• så hun må hente ...
  In Danish, a comma is standard before a conjunction like så when it introduces a new clause like this.

Here så is a conjunction meaning so / therefore / as a result:

• ..., så hun må hente dem igen. = “..., so she has to fetch them again.”

Danish så can also be an adverb meaning so/very (degree):

• Det er så godt. = “It’s so good.”
  Different function, same word.

Both can occur, but they have different structures:

• ..., så hun må hente dem igen. → så introduces a full clause with normal main-clause order (subject hun before the finite verb må).
• ..., så må hun hente dem igen. → here så acts more like an adverb at the start of a main clause, which triggers inversion (finite verb må before subject hun).

In practice, both can be translated similarly, but the grammar behind the word order differs.

In this context må means must / have to (necessity):

• hun må hente dem igen = she has to fetch them again.

må can also mean may / be allowed to depending on context:

• Du må gå nu. = You may go now / You’re allowed to go now.
  Here, the “have to” reading is supported by the cause-and-effect setup: she forgot them, so she needs to retrieve them.

dem is the object form (“them”), while de is the subject form (“they”).

• Subject: De er på bordet. = They are on the table.
• Object: Hun henter dem. = She fetches them.

So dem refers back to piller as the object of hente.

igen means again. In your sentence it comes at the end:

• ... hente dem igen = fetch them again

It can sometimes appear earlier for emphasis or style, but end position is very common and natural with this meaning.

bordet is the definite form: the table.

• Indefinite: et bord = a table
• Definite: bordet = the table

Danish often uses the definite form when English might also use the, especially when the table is understood from context (e.g., the kitchen table).

Yes, glemmer is present tense. Danish present tense often uses -r:

• infinitive: at glemme
• present: glemmer
• past: glemte
• past participle: glemt

The tense depends on context; this sentence can describe a habitual situation (“She forgets...”) or a vivid present (“She forgets... so she has to...”).

It follows standard Danish main-clause order: Subject – Verb – (Object) – (Adverbials):

• Hun (subject)
• glemmer (finite verb)
• sine piller (object)
• på bordet (place adverbial)

Yes. It changes definiteness/meaning nuance:

• sine piller focuses on whose pills they are (her own).
• pillerne = the pills, typically already known in context.

If it’s already clear they’re hers, pillerne can sound more natural; if you want to make ownership explicit, sine piller does that.